Theorem wemaplem2 9519

Description: Lemma for wemapso 9523. Transitivity. (Contributed by Stefan O'Rear, 17-Jan-2015.) (Revised by AV, 21-Jul-2024.)

Hypotheses

Ref	Expression
wemapso.t	⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))}
wemaplem2.p	⊢ (𝜑 → 𝑃 ∈ (𝐵 ↑m 𝐴))
wemaplem2.x	⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m 𝐴))
wemaplem2.q	⊢ (𝜑 → 𝑄 ∈ (𝐵 ↑m 𝐴))
wemaplem2.r	⊢ (𝜑 → 𝑅 Or 𝐴)
wemaplem2.s	⊢ (𝜑 → 𝑆 Po 𝐵)
wemaplem2.px1	⊢ (𝜑 → 𝑎 ∈ 𝐴)
wemaplem2.px2	⊢ (𝜑 → (𝑃‘𝑎)𝑆(𝑋‘𝑎))
wemaplem2.px3	⊢ (𝜑 → ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐)))
wemaplem2.xq1	⊢ (𝜑 → 𝑏 ∈ 𝐴)
wemaplem2.xq2	⊢ (𝜑 → (𝑋‘𝑏)𝑆(𝑄‘𝑏))
wemaplem2.xq3	⊢ (𝜑 → ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐)))

Assertion

Ref	Expression
wemaplem2	⊢ (𝜑 → 𝑃𝑇𝑄)

Distinct variable groups:   𝑎,𝑏,𝑐,𝑥,𝐵   𝑇,𝑎,𝑏,𝑐   𝑤,𝑎,𝑦,𝑧,𝑋,𝑏,𝑐,𝑥   𝐴,𝑎,𝑏,𝑐,𝑤,𝑥,𝑦,𝑧   𝑃,𝑎,𝑏,𝑐,𝑤,𝑥,𝑦,𝑧   𝑄,𝑎,𝑏,𝑐,𝑤,𝑥,𝑦,𝑧   𝑅,𝑎,𝑏,𝑐,𝑤,𝑥,𝑦,𝑧   𝑆,𝑎,𝑏,𝑐,𝑤,𝑥,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥, 𝑦, 𝑧, 𝑤, 𝑎, 𝑏, 𝑐)   𝐵(𝑦, 𝑧, 𝑤)   𝑇(𝑥, 𝑦, 𝑧, 𝑤)

Proof of Theorem wemaplem2

Dummy variable 𝑑 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wemaplem2.px1	. . . 4 ⊢ (𝜑 → 𝑎 ∈ 𝐴)
2		wemaplem2.xq1	. . . 4 ⊢ (𝜑 → 𝑏 ∈ 𝐴)
3	1, 2	ifcld 4539	. . 3 ⊢ (𝜑 → if(𝑎𝑅𝑏, 𝑎, 𝑏) ∈ 𝐴)
4		wemaplem2.px2	. . . . . . 7 ⊢ (𝜑 → (𝑃‘𝑎)𝑆(𝑋‘𝑎))
5	4	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎𝑅𝑏) → (𝑃‘𝑎)𝑆(𝑋‘𝑎))
6		breq1 5117	. . . . . . . . 9 ⊢ (𝑐 = 𝑎 → (𝑐𝑅𝑏 ↔ 𝑎𝑅𝑏))
7		fveq2 6888	. . . . . . . . . 10 ⊢ (𝑐 = 𝑎 → (𝑋‘𝑐) = (𝑋‘𝑎))
8		fveq2 6888	. . . . . . . . . 10 ⊢ (𝑐 = 𝑎 → (𝑄‘𝑐) = (𝑄‘𝑎))
9	7, 8	eqeq12d 2782	. . . . . . . . 9 ⊢ (𝑐 = 𝑎 → ((𝑋‘𝑐) = (𝑄‘𝑐) ↔ (𝑋‘𝑎) = (𝑄‘𝑎)))
10	6, 9	imbi12d 347	. . . . . . . 8 ⊢ (𝑐 = 𝑎 → ((𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐)) ↔ (𝑎𝑅𝑏 → (𝑋‘𝑎) = (𝑄‘𝑎))))
11		wemaplem2.xq3	. . . . . . . 8 ⊢ (𝜑 → ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐)))
12	10, 11, 1	rspcdva 3585	. . . . . . 7 ⊢ (𝜑 → (𝑎𝑅𝑏 → (𝑋‘𝑎) = (𝑄‘𝑎)))
13	12	imp 412	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎𝑅𝑏) → (𝑋‘𝑎) = (𝑄‘𝑎))
14	5, 13	breqtrd 5142	. . . . 5 ⊢ ((𝜑 ∧ 𝑎𝑅𝑏) → (𝑃‘𝑎)𝑆(𝑄‘𝑎))
15		iftrue 4498	. . . . . . . 8 ⊢ (𝑎𝑅𝑏 → if(𝑎𝑅𝑏, 𝑎, 𝑏) = 𝑎)
16	15	fveq2d 6892	. . . . . . 7 ⊢ (𝑎𝑅𝑏 → (𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) = (𝑃‘𝑎))
17	15	fveq2d 6892	. . . . . . 7 ⊢ (𝑎𝑅𝑏 → (𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) = (𝑄‘𝑎))
18	16, 17	breq12d 5127	. . . . . 6 ⊢ (𝑎𝑅𝑏 → ((𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) ↔ (𝑃‘𝑎)𝑆(𝑄‘𝑎)))
19	18	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑎𝑅𝑏) → ((𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) ↔ (𝑃‘𝑎)𝑆(𝑄‘𝑎)))
20	14, 19	mpbird 260	. . . 4 ⊢ ((𝜑 ∧ 𝑎𝑅𝑏) → (𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)))
21		wemaplem2.s	. . . . . . 7 ⊢ (𝜑 → 𝑆 Po 𝐵)
22	21	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 = 𝑏) → 𝑆 Po 𝐵)
23		wemaplem2.p	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ (𝐵 ↑m 𝐴))
24		elmapi 8855	. . . . . . . . . 10 ⊢ (𝑃 ∈ (𝐵 ↑m 𝐴) → 𝑃:𝐴⟶𝐵)
25	23, 24	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑃:𝐴⟶𝐵)
26	25, 2	ffvelcdmd 7087	. . . . . . . 8 ⊢ (𝜑 → (𝑃‘𝑏) ∈ 𝐵)
27		wemaplem2.x	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m 𝐴))
28		elmapi 8855	. . . . . . . . . 10 ⊢ (𝑋 ∈ (𝐵 ↑m 𝐴) → 𝑋:𝐴⟶𝐵)
29	27, 28	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑋:𝐴⟶𝐵)
30	29, 2	ffvelcdmd 7087	. . . . . . . 8 ⊢ (𝜑 → (𝑋‘𝑏) ∈ 𝐵)
31		wemaplem2.q	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ∈ (𝐵 ↑m 𝐴))
32		elmapi 8855	. . . . . . . . . 10 ⊢ (𝑄 ∈ (𝐵 ↑m 𝐴) → 𝑄:𝐴⟶𝐵)
33	31, 32	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑄:𝐴⟶𝐵)
34	33, 2	ffvelcdmd 7087	. . . . . . . 8 ⊢ (𝜑 → (𝑄‘𝑏) ∈ 𝐵)
35	26, 30, 34	3jca 1146	. . . . . . 7 ⊢ (𝜑 → ((𝑃‘𝑏) ∈ 𝐵 ∧ (𝑋‘𝑏) ∈ 𝐵 ∧ (𝑄‘𝑏) ∈ 𝐵))
36	35	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 = 𝑏) → ((𝑃‘𝑏) ∈ 𝐵 ∧ (𝑋‘𝑏) ∈ 𝐵 ∧ (𝑄‘𝑏) ∈ 𝐵))
37		fveq2 6888	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → (𝑃‘𝑎) = (𝑃‘𝑏))
38		fveq2 6888	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → (𝑋‘𝑎) = (𝑋‘𝑏))
39	37, 38	breq12d 5127	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ↔ (𝑃‘𝑏)𝑆(𝑋‘𝑏)))
40	4, 39	syl5ibcom 248	. . . . . . 7 ⊢ (𝜑 → (𝑎 = 𝑏 → (𝑃‘𝑏)𝑆(𝑋‘𝑏)))
41	40	imp 412	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 = 𝑏) → (𝑃‘𝑏)𝑆(𝑋‘𝑏))
42		wemaplem2.xq2	. . . . . . 7 ⊢ (𝜑 → (𝑋‘𝑏)𝑆(𝑄‘𝑏))
43	42	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 = 𝑏) → (𝑋‘𝑏)𝑆(𝑄‘𝑏))
44		potr 5587	. . . . . . 7 ⊢ ((𝑆 Po 𝐵 ∧ ((𝑃‘𝑏) ∈ 𝐵 ∧ (𝑋‘𝑏) ∈ 𝐵 ∧ (𝑄‘𝑏) ∈ 𝐵)) → (((𝑃‘𝑏)𝑆(𝑋‘𝑏) ∧ (𝑋‘𝑏)𝑆(𝑄‘𝑏)) → (𝑃‘𝑏)𝑆(𝑄‘𝑏)))
45	44	imp 412	. . . . . 6 ⊢ (((𝑆 Po 𝐵 ∧ ((𝑃‘𝑏) ∈ 𝐵 ∧ (𝑋‘𝑏) ∈ 𝐵 ∧ (𝑄‘𝑏) ∈ 𝐵)) ∧ ((𝑃‘𝑏)𝑆(𝑋‘𝑏) ∧ (𝑋‘𝑏)𝑆(𝑄‘𝑏))) → (𝑃‘𝑏)𝑆(𝑄‘𝑏))
46	22, 36, 41, 43, 45	syl22anc 852	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 = 𝑏) → (𝑃‘𝑏)𝑆(𝑄‘𝑏))
47		ifeq1 4496	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → if(𝑎𝑅𝑏, 𝑎, 𝑏) = if(𝑎𝑅𝑏, 𝑏, 𝑏))
48		ifid 4533	. . . . . . . . 9 ⊢ if(𝑎𝑅𝑏, 𝑏, 𝑏) = 𝑏
49	47, 48	eqtrdi 2817	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → if(𝑎𝑅𝑏, 𝑎, 𝑏) = 𝑏)
50	49	fveq2d 6892	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) = (𝑃‘𝑏))
51	49	fveq2d 6892	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) = (𝑄‘𝑏))
52	50, 51	breq12d 5127	. . . . . 6 ⊢ (𝑎 = 𝑏 → ((𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) ↔ (𝑃‘𝑏)𝑆(𝑄‘𝑏)))
53	52	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 = 𝑏) → ((𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) ↔ (𝑃‘𝑏)𝑆(𝑄‘𝑏)))
54	46, 53	mpbird 260	. . . 4 ⊢ ((𝜑 ∧ 𝑎 = 𝑏) → (𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)))
55		breq1 5117	. . . . . . . . 9 ⊢ (𝑐 = 𝑏 → (𝑐𝑅𝑎 ↔ 𝑏𝑅𝑎))
56		fveq2 6888	. . . . . . . . . 10 ⊢ (𝑐 = 𝑏 → (𝑃‘𝑐) = (𝑃‘𝑏))
57		fveq2 6888	. . . . . . . . . 10 ⊢ (𝑐 = 𝑏 → (𝑋‘𝑐) = (𝑋‘𝑏))
58	56, 57	eqeq12d 2782	. . . . . . . . 9 ⊢ (𝑐 = 𝑏 → ((𝑃‘𝑐) = (𝑋‘𝑐) ↔ (𝑃‘𝑏) = (𝑋‘𝑏)))
59	55, 58	imbi12d 347	. . . . . . . 8 ⊢ (𝑐 = 𝑏 → ((𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐)) ↔ (𝑏𝑅𝑎 → (𝑃‘𝑏) = (𝑋‘𝑏))))
60		wemaplem2.px3	. . . . . . . 8 ⊢ (𝜑 → ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐)))
61	59, 60, 2	rspcdva 3585	. . . . . . 7 ⊢ (𝜑 → (𝑏𝑅𝑎 → (𝑃‘𝑏) = (𝑋‘𝑏)))
62	61	imp 412	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏𝑅𝑎) → (𝑃‘𝑏) = (𝑋‘𝑏))
63	42	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏𝑅𝑎) → (𝑋‘𝑏)𝑆(𝑄‘𝑏))
64	62, 63	eqbrtrd 5138	. . . . 5 ⊢ ((𝜑 ∧ 𝑏𝑅𝑎) → (𝑃‘𝑏)𝑆(𝑄‘𝑏))
65		wemaplem2.r	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 Or 𝐴)
66		sopo 5593	. . . . . . . . 9 ⊢ (𝑅 Or 𝐴 → 𝑅 Po 𝐴)
67	65, 66	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑅 Po 𝐴)
68		po2nr 5588	. . . . . . . 8 ⊢ ((𝑅 Po 𝐴 ∧ (𝑏 ∈ 𝐴 ∧ 𝑎 ∈ 𝐴)) → ¬ (𝑏𝑅𝑎 ∧ 𝑎𝑅𝑏))
69	67, 2, 1, 68	syl12anc 850	. . . . . . 7 ⊢ (𝜑 → ¬ (𝑏𝑅𝑎 ∧ 𝑎𝑅𝑏))
70		nan 843	. . . . . . 7 ⊢ ((𝜑 → ¬ (𝑏𝑅𝑎 ∧ 𝑎𝑅𝑏)) ↔ ((𝜑 ∧ 𝑏𝑅𝑎) → ¬ 𝑎𝑅𝑏))
71	69, 70	mpbi 233	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏𝑅𝑎) → ¬ 𝑎𝑅𝑏)
72		iffalse 4501	. . . . . . . 8 ⊢ (¬ 𝑎𝑅𝑏 → if(𝑎𝑅𝑏, 𝑎, 𝑏) = 𝑏)
73	72	fveq2d 6892	. . . . . . 7 ⊢ (¬ 𝑎𝑅𝑏 → (𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) = (𝑃‘𝑏))
74	72	fveq2d 6892	. . . . . . 7 ⊢ (¬ 𝑎𝑅𝑏 → (𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) = (𝑄‘𝑏))
75	73, 74	breq12d 5127	. . . . . 6 ⊢ (¬ 𝑎𝑅𝑏 → ((𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) ↔ (𝑃‘𝑏)𝑆(𝑄‘𝑏)))
76	71, 75	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑏𝑅𝑎) → ((𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) ↔ (𝑃‘𝑏)𝑆(𝑄‘𝑏)))
77	64, 76	mpbird 260	. . . 4 ⊢ ((𝜑 ∧ 𝑏𝑅𝑎) → (𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)))
78		solin 5601	. . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → (𝑎𝑅𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑅𝑎))
79	65, 1, 2, 78	syl12anc 850	. . . 4 ⊢ (𝜑 → (𝑎𝑅𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑅𝑎))
80	20, 54, 77, 79	mpjao3dan 1459	. . 3 ⊢ (𝜑 → (𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)))
81		r19.26 3128	. . . . 5 ⊢ (∀𝑐 ∈ 𝐴 ((𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐)) ∧ (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))) ↔ (∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐)) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))))
82	60, 11, 81	sylanbrc 595	. . . 4 ⊢ (𝜑 → ∀𝑐 ∈ 𝐴 ((𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐)) ∧ (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))))
83	65, 1, 2	3jca 1146	. . . . 5 ⊢ (𝜑 → (𝑅 Or 𝐴 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴))
84		anim12 821	. . . . . . 7 ⊢ (((𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐)) ∧ (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))) → ((𝑐𝑅𝑎 ∧ 𝑐𝑅𝑏) → ((𝑃‘𝑐) = (𝑋‘𝑐) ∧ (𝑋‘𝑐) = (𝑄‘𝑐))))
85		eqtr 2786	. . . . . . 7 ⊢ (((𝑃‘𝑐) = (𝑋‘𝑐) ∧ (𝑋‘𝑐) = (𝑄‘𝑐)) → (𝑃‘𝑐) = (𝑄‘𝑐))
86	84, 85	syl6 36	. . . . . 6 ⊢ (((𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐)) ∧ (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))) → ((𝑐𝑅𝑎 ∧ 𝑐𝑅𝑏) → (𝑃‘𝑐) = (𝑄‘𝑐)))
87	86	ralimi 3105	. . . . 5 ⊢ (∀𝑐 ∈ 𝐴 ((𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐)) ∧ (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))) → ∀𝑐 ∈ 𝐴 ((𝑐𝑅𝑎 ∧ 𝑐𝑅𝑏) → (𝑃‘𝑐) = (𝑄‘𝑐)))
88		simpl1 1210	. . . . . . . . 9 ⊢ (((𝑅 Or 𝐴 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ 𝐴) → 𝑅 Or 𝐴)
89		simpr 490	. . . . . . . . 9 ⊢ (((𝑅 Or 𝐴 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ 𝐴) → 𝑐 ∈ 𝐴)
90		simpl2 1211	. . . . . . . . 9 ⊢ (((𝑅 Or 𝐴 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ 𝐴) → 𝑎 ∈ 𝐴)
91		simpl3 1212	. . . . . . . . 9 ⊢ (((𝑅 Or 𝐴 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ 𝐴) → 𝑏 ∈ 𝐴)
92		soltmin 6141	. . . . . . . . 9 ⊢ ((𝑅 Or 𝐴 ∧ (𝑐 ∈ 𝐴 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → (𝑐𝑅if(𝑎𝑅𝑏, 𝑎, 𝑏) ↔ (𝑐𝑅𝑎 ∧ 𝑐𝑅𝑏)))
93	88, 89, 90, 91, 92	syl13anc 1399	. . . . . . . 8 ⊢ (((𝑅 Or 𝐴 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ 𝐴) → (𝑐𝑅if(𝑎𝑅𝑏, 𝑎, 𝑏) ↔ (𝑐𝑅𝑎 ∧ 𝑐𝑅𝑏)))
94	93	biimpd 232	. . . . . . 7 ⊢ (((𝑅 Or 𝐴 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ 𝐴) → (𝑐𝑅if(𝑎𝑅𝑏, 𝑎, 𝑏) → (𝑐𝑅𝑎 ∧ 𝑐𝑅𝑏)))
95	94	imim1d 83	. . . . . 6 ⊢ (((𝑅 Or 𝐴 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ 𝐴) → (((𝑐𝑅𝑎 ∧ 𝑐𝑅𝑏) → (𝑃‘𝑐) = (𝑄‘𝑐)) → (𝑐𝑅if(𝑎𝑅𝑏, 𝑎, 𝑏) → (𝑃‘𝑐) = (𝑄‘𝑐))))
96	95	ralimdva 3180	. . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → (∀𝑐 ∈ 𝐴 ((𝑐𝑅𝑎 ∧ 𝑐𝑅𝑏) → (𝑃‘𝑐) = (𝑄‘𝑐)) → ∀𝑐 ∈ 𝐴 (𝑐𝑅if(𝑎𝑅𝑏, 𝑎, 𝑏) → (𝑃‘𝑐) = (𝑄‘𝑐))))
97	83, 87, 96	syl2im 41	. . . 4 ⊢ (𝜑 → (∀𝑐 ∈ 𝐴 ((𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐)) ∧ (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))) → ∀𝑐 ∈ 𝐴 (𝑐𝑅if(𝑎𝑅𝑏, 𝑎, 𝑏) → (𝑃‘𝑐) = (𝑄‘𝑐))))
98	82, 97	mpd 16	. . 3 ⊢ (𝜑 → ∀𝑐 ∈ 𝐴 (𝑐𝑅if(𝑎𝑅𝑏, 𝑎, 𝑏) → (𝑃‘𝑐) = (𝑄‘𝑐)))
99		fveq2 6888	. . . . . 6 ⊢ (𝑑 = if(𝑎𝑅𝑏, 𝑎, 𝑏) → (𝑃‘𝑑) = (𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏)))
100		fveq2 6888	. . . . . 6 ⊢ (𝑑 = if(𝑎𝑅𝑏, 𝑎, 𝑏) → (𝑄‘𝑑) = (𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)))
101	99, 100	breq12d 5127	. . . . 5 ⊢ (𝑑 = if(𝑎𝑅𝑏, 𝑎, 𝑏) → ((𝑃‘𝑑)𝑆(𝑄‘𝑑) ↔ (𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏))))
102		breq2 5118	. . . . . . 7 ⊢ (𝑑 = if(𝑎𝑅𝑏, 𝑎, 𝑏) → (𝑐𝑅𝑑 ↔ 𝑐𝑅if(𝑎𝑅𝑏, 𝑎, 𝑏)))
103	102	imbi1d 344	. . . . . 6 ⊢ (𝑑 = if(𝑎𝑅𝑏, 𝑎, 𝑏) → ((𝑐𝑅𝑑 → (𝑃‘𝑐) = (𝑄‘𝑐)) ↔ (𝑐𝑅if(𝑎𝑅𝑏, 𝑎, 𝑏) → (𝑃‘𝑐) = (𝑄‘𝑐))))
104	103	ralbidv 3191	. . . . 5 ⊢ (𝑑 = if(𝑎𝑅𝑏, 𝑎, 𝑏) → (∀𝑐 ∈ 𝐴 (𝑐𝑅𝑑 → (𝑃‘𝑐) = (𝑄‘𝑐)) ↔ ∀𝑐 ∈ 𝐴 (𝑐𝑅if(𝑎𝑅𝑏, 𝑎, 𝑏) → (𝑃‘𝑐) = (𝑄‘𝑐))))
105	101, 104	anbi12d 644	. . . 4 ⊢ (𝑑 = if(𝑎𝑅𝑏, 𝑎, 𝑏) → (((𝑃‘𝑑)𝑆(𝑄‘𝑑) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑑 → (𝑃‘𝑐) = (𝑄‘𝑐))) ↔ ((𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅if(𝑎𝑅𝑏, 𝑎, 𝑏) → (𝑃‘𝑐) = (𝑄‘𝑐)))))
106	105	rspcev 3584	. . 3 ⊢ ((if(𝑎𝑅𝑏, 𝑎, 𝑏) ∈ 𝐴 ∧ ((𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅if(𝑎𝑅𝑏, 𝑎, 𝑏) → (𝑃‘𝑐) = (𝑄‘𝑐)))) → ∃𝑑 ∈ 𝐴 ((𝑃‘𝑑)𝑆(𝑄‘𝑑) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑑 → (𝑃‘𝑐) = (𝑄‘𝑐))))
107	3, 80, 98, 106	syl12anc 850	. 2 ⊢ (𝜑 → ∃𝑑 ∈ 𝐴 ((𝑃‘𝑑)𝑆(𝑄‘𝑑) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑑 → (𝑃‘𝑐) = (𝑄‘𝑐))))
108		wemapso.t	. . . 4 ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))}
109	108	wemaplem1 9518	. . 3 ⊢ ((𝑃 ∈ (𝐵 ↑m 𝐴) ∧ 𝑄 ∈ (𝐵 ↑m 𝐴)) → (𝑃𝑇𝑄 ↔ ∃𝑑 ∈ 𝐴 ((𝑃‘𝑑)𝑆(𝑄‘𝑑) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑑 → (𝑃‘𝑐) = (𝑄‘𝑐)))))
110	23, 31, 109	syl2anc 596	. 2 ⊢ (𝜑 → (𝑃𝑇𝑄 ↔ ∃𝑑 ∈ 𝐴 ((𝑃‘𝑑)𝑆(𝑄‘𝑑) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑑 → (𝑃‘𝑐) = (𝑄‘𝑐)))))
111	107, 110	mpbird 260	1 ⊢ (𝜑 → 𝑃𝑇𝑄)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ w3o 1102   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3082  ∃wrex 3092  ifcif 4492   class class class wbr 5114  {copab 5178   Po wpo 5572   Or wor 5573  ⟶wf 6539  ‘cfv 6543  (class class class)co 7423   ↑_m cmap 8833

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-po 5574  df-so 5575  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7426  df-oprab 7427  df-mpo 7428  df-1st 7995  df-2nd 7996  df-map 8835

This theorem is used by:  wemaplem3  9520