Theorem wemaplem3 9508

Description: Lemma for wemapso 9511. 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 9506	. . . 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 584	. . 3 ⊢ (𝜑 → (𝑃𝑇𝑋 ↔ ∃𝑎 ∈ 𝐴 ((𝑃‘𝑎)𝑆(𝑋‘𝑎) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐)))))
7	1, 6	mpbid 232	. 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 9506	. . . 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 584	. . 3 ⊢ (𝜑 → (𝑋𝑇𝑄 ↔ ∃𝑏 ∈ 𝐴 ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐)))))
12	8, 11	mpbid 232	. 2 ⊢ (𝜑 → ∃𝑏 ∈ 𝐴 ((𝑋‘𝑏)𝑆(𝑄‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))))
13	2	ad2antrr 726	. . . . 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 726	. . . . 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 726	. . . . 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 726	. . . . 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 726	. . . . 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 776	. . . . 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 1243	. . . . . 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 1118	. . . . 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 772	. . . . . 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 727	. . . . 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 770	. . . . 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 780	. . . . 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 781	. . . . 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 9507	. . . 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 3136	. . 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 3136	. 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 49	1 ⊢ (𝜑 → 𝑃𝑇𝑄)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054   class class class wbr 5110  {copab 5172   Po wpo 5547   Or wor 5548  ‘cfv 6514  (class class class)co 7390   ↑_m cmap 8802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-po 5549  df-so 5550  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-map 8804

This theorem is referenced by:  wemappo  9509