Theorem wemaplem3 9523

Description: Lemma for wemapso 9526. 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)
wemaplem3.px	⊢ (𝜑 → 𝑃𝑇𝑋)
wemaplem3.xq	⊢ (𝜑 → 𝑋𝑇𝑄)

Assertion

Ref	Expression
wemaplem3	⊢ (𝜑 → 𝑃𝑇𝑄)

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

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

Proof of Theorem wemaplem3

Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		wemaplem3.px	. . 3 ⊢ (𝜑 → 𝑃𝑇𝑋)
2		wemaplem2.p	. . . 4 ⊢ (𝜑 → 𝑃 ∈ (𝐵 ↑m 𝐴))
3		wemaplem2.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m 𝐴))
4		wemapso.t	. . . . 5 ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))}
5	4	wemaplem1 9521	. . . 4 ⊢ ((𝑃 ∈ (𝐵 ↑m 𝐴) ∧ 𝑋 ∈ (𝐵 ↑m 𝐴)) → (𝑃𝑇𝑋 ↔ ∃𝑎 ∈ 𝐴 ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐)))))
6	2, 3, 5	syl2anc 596	. . 3 ⊢ (𝜑 → (𝑃𝑇𝑋 ↔ ∃𝑎 ∈ 𝐴 ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐)))))
7	1, 6	mpbid 235	. 2 ⊢ (𝜑 → ∃𝑎 ∈ 𝐴 ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))))
8		wemaplem3.xq	. . 3 ⊢ (𝜑 → 𝑋𝑇𝑄)
9		wemaplem2.q	. . . 4 ⊢ (𝜑 → 𝑄 ∈ (𝐵 ↑m 𝐴))
10	4	wemaplem1 9521	. . . 4 ⊢ ((𝑋 ∈ (𝐵 ↑m 𝐴) ∧ 𝑄 ∈ (𝐵 ↑m 𝐴)) → (𝑋𝑇𝑄 ↔ ∃𝑏 ∈ 𝐴 ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐)))))
11	3, 9, 10	syl2anc 596	. . 3 ⊢ (𝜑 → (𝑋𝑇𝑄 ↔ ∃𝑏 ∈ 𝐴 ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐)))))
12	8, 11	mpbid 235	. 2 ⊢ (𝜑 → ∃𝑏 ∈ 𝐴 ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))))
13	2	ad2antrr 739	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))))) ∧ (𝑏 ∈ 𝐴 ∧ ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))))) → 𝑃 ∈ (𝐵 ↑m 𝐴))
14	3	ad2antrr 739	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))))) ∧ (𝑏 ∈ 𝐴 ∧ ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))))) → 𝑋 ∈ (𝐵 ↑m 𝐴))
15	9	ad2antrr 739	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))))) ∧ (𝑏 ∈ 𝐴 ∧ ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))))) → 𝑄 ∈ (𝐵 ↑m 𝐴))
16		wemaplem2.r	. . . . . 6 ⊢ (𝜑 → 𝑅 Or 𝐴)
17	16	ad2antrr 739	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))))) ∧ (𝑏 ∈ 𝐴 ∧ ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))))) → 𝑅 Or 𝐴)
18		wemaplem2.s	. . . . . 6 ⊢ (𝜑 → 𝑆 Po 𝐵)
19	18	ad2antrr 739	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))))) ∧ (𝑏 ∈ 𝐴 ∧ ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))))) → 𝑆 Po 𝐵)
20		simplrl 789	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))))) ∧ (𝑏 ∈ 𝐴 ∧ ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))))) → 𝑎 ∈ 𝐴)
21		simp2rl 1261	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐)))) ∧ (𝑏 ∈ 𝐴 ∧ ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))))) → (𝑃‘𝑎)𝑆(𝑋‘𝑎))
22	21	3expa 1136	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))))) ∧ (𝑏 ∈ 𝐴 ∧ ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))))) → (𝑃‘𝑎)𝑆(𝑋‘𝑎))
23		simprr 785	. . . . . 6 ⊢ ((𝑎 ∈ 𝐴 ∧ ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐)))) → ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐)))
24	23	ad2antlr 740	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))))) ∧ (𝑏 ∈ 𝐴 ∧ ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))))) → ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐)))
25		simprl 783	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))))) ∧ (𝑏 ∈ 𝐴 ∧ ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))))) → 𝑏 ∈ 𝐴)
26		simprrl 793	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))))) ∧ (𝑏 ∈ 𝐴 ∧ ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))))) → (𝑋‘𝑏)𝑆(𝑄‘𝑏))
27		simprrr 794	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))))) ∧ (𝑏 ∈ 𝐴 ∧ ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))))) → ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐)))
28	4, 13, 14, 15, 17, 19, 20, 22, 24, 25, 26, 27	wemaplem2 9522	. . . 4 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))))) ∧ (𝑏 ∈ 𝐴 ∧ ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))))) → 𝑃𝑇𝑄)
29	28	rexlimdvaa 3164	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))))) → (∃𝑏 ∈ 𝐴 ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))) → 𝑃𝑇𝑄))
30	29	rexlimdvaa 3164	. 2 ⊢ (𝜑 → (∃𝑎 ∈ 𝐴 ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))) → (∃𝑏 ∈ 𝐴 ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))) → 𝑃𝑇𝑄)))
31	7, 12, 30	mp2d 50	1 ⊢ (𝜑 → 𝑃𝑇𝑄)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ∃wrex 3086   class class class wbr 5103  {copab 5167   Po wpo 5561   Or wor 5562  ‘cfv 6533  (class class class)co 7414   ↑_m cmap 8829

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-po 5563  df-so 5564  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-map 8831

This theorem is used by:  wemappo  9524