Theorem fvcofneq 6631

Description: The values of two function compositions are equal if the values of the composed functions are pairwise equal. (Contributed by AV, 26-Jan-2019.)

Assertion

Ref	Expression
fvcofneq	⊢ ((𝐺 Fn 𝐴 ∧ 𝐾 Fn 𝐵) → ((𝑋 ∈ (𝐴 ∩ 𝐵) ∧ (𝐺‘𝑋) = (𝐾‘𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹‘𝑥) = (𝐻‘𝑥)) → ((𝐹 ∘ 𝐺)‘𝑋) = ((𝐻 ∘ 𝐾)‘𝑋)))

Distinct variable groups:   𝑥,𝐹   𝑥,𝐺   𝑥,𝐻   𝑥,𝐾   𝑥,𝑋

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem fvcofneq

Step	Hyp	Ref	Expression
1		simpl 476	. . . 4 ⊢ ((𝐺 Fn 𝐴 ∧ 𝐾 Fn 𝐵) → 𝐺 Fn 𝐴)
2		elinel1 4022	. . . . 5 ⊢ (𝑋 ∈ (𝐴 ∩ 𝐵) → 𝑋 ∈ 𝐴)
3	2	3ad2ant1 1124	. . . 4 ⊢ ((𝑋 ∈ (𝐴 ∩ 𝐵) ∧ (𝐺‘𝑋) = (𝐾‘𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹‘𝑥) = (𝐻‘𝑥)) → 𝑋 ∈ 𝐴)
4		fvco2 6533	. . . 4 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋)))
5	1, 3, 4	syl2an 589	. . 3 ⊢ (((𝐺 Fn 𝐴 ∧ 𝐾 Fn 𝐵) ∧ (𝑋 ∈ (𝐴 ∩ 𝐵) ∧ (𝐺‘𝑋) = (𝐾‘𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹‘𝑥) = (𝐻‘𝑥))) → ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋)))
6		simpr 479	. . . . 5 ⊢ ((𝐺 Fn 𝐴 ∧ 𝐾 Fn 𝐵) → 𝐾 Fn 𝐵)
7		elinel2 4023	. . . . . 6 ⊢ (𝑋 ∈ (𝐴 ∩ 𝐵) → 𝑋 ∈ 𝐵)
8	7	3ad2ant1 1124	. . . . 5 ⊢ ((𝑋 ∈ (𝐴 ∩ 𝐵) ∧ (𝐺‘𝑋) = (𝐾‘𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹‘𝑥) = (𝐻‘𝑥)) → 𝑋 ∈ 𝐵)
9		fvco2 6533	. . . . 5 ⊢ ((𝐾 Fn 𝐵 ∧ 𝑋 ∈ 𝐵) → ((𝐻 ∘ 𝐾)‘𝑋) = (𝐻‘(𝐾‘𝑋)))
10	6, 8, 9	syl2an 589	. . . 4 ⊢ (((𝐺 Fn 𝐴 ∧ 𝐾 Fn 𝐵) ∧ (𝑋 ∈ (𝐴 ∩ 𝐵) ∧ (𝐺‘𝑋) = (𝐾‘𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹‘𝑥) = (𝐻‘𝑥))) → ((𝐻 ∘ 𝐾)‘𝑋) = (𝐻‘(𝐾‘𝑋)))
11		fveq2 6446	. . . . . . 7 ⊢ ((𝐾‘𝑋) = (𝐺‘𝑋) → (𝐻‘(𝐾‘𝑋)) = (𝐻‘(𝐺‘𝑋)))
12	11	eqcoms 2786	. . . . . 6 ⊢ ((𝐺‘𝑋) = (𝐾‘𝑋) → (𝐻‘(𝐾‘𝑋)) = (𝐻‘(𝐺‘𝑋)))
13	12	3ad2ant2 1125	. . . . 5 ⊢ ((𝑋 ∈ (𝐴 ∩ 𝐵) ∧ (𝐺‘𝑋) = (𝐾‘𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹‘𝑥) = (𝐻‘𝑥)) → (𝐻‘(𝐾‘𝑋)) = (𝐻‘(𝐺‘𝑋)))
14	13	adantl 475	. . . 4 ⊢ (((𝐺 Fn 𝐴 ∧ 𝐾 Fn 𝐵) ∧ (𝑋 ∈ (𝐴 ∩ 𝐵) ∧ (𝐺‘𝑋) = (𝐾‘𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹‘𝑥) = (𝐻‘𝑥))) → (𝐻‘(𝐾‘𝑋)) = (𝐻‘(𝐺‘𝑋)))
15		id 22	. . . . . . . . . . . 12 ⊢ (𝐺 Fn 𝐴 → 𝐺 Fn 𝐴)
16		fnfvelrn 6620	. . . . . . . . . . . 12 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐺‘𝑋) ∈ ran 𝐺)
17	15, 2, 16	syl2anr 590	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ (𝐴 ∩ 𝐵) ∧ 𝐺 Fn 𝐴) → (𝐺‘𝑋) ∈ ran 𝐺)
18	17	ex 403	. . . . . . . . . 10 ⊢ (𝑋 ∈ (𝐴 ∩ 𝐵) → (𝐺 Fn 𝐴 → (𝐺‘𝑋) ∈ ran 𝐺))
19		id 22	. . . . . . . . . . . 12 ⊢ (𝐾 Fn 𝐵 → 𝐾 Fn 𝐵)
20		fnfvelrn 6620	. . . . . . . . . . . 12 ⊢ ((𝐾 Fn 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝐾‘𝑋) ∈ ran 𝐾)
21	19, 7, 20	syl2anr 590	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ (𝐴 ∩ 𝐵) ∧ 𝐾 Fn 𝐵) → (𝐾‘𝑋) ∈ ran 𝐾)
22	21	ex 403	. . . . . . . . . 10 ⊢ (𝑋 ∈ (𝐴 ∩ 𝐵) → (𝐾 Fn 𝐵 → (𝐾‘𝑋) ∈ ran 𝐾))
23	18, 22	anim12d 602	. . . . . . . . 9 ⊢ (𝑋 ∈ (𝐴 ∩ 𝐵) → ((𝐺 Fn 𝐴 ∧ 𝐾 Fn 𝐵) → ((𝐺‘𝑋) ∈ ran 𝐺 ∧ (𝐾‘𝑋) ∈ ran 𝐾)))
24		eleq1 2847	. . . . . . . . . . . 12 ⊢ ((𝐾‘𝑋) = (𝐺‘𝑋) → ((𝐾‘𝑋) ∈ ran 𝐾 ↔ (𝐺‘𝑋) ∈ ran 𝐾))
25	24	eqcoms 2786	. . . . . . . . . . 11 ⊢ ((𝐺‘𝑋) = (𝐾‘𝑋) → ((𝐾‘𝑋) ∈ ran 𝐾 ↔ (𝐺‘𝑋) ∈ ran 𝐾))
26	25	anbi2d 622	. . . . . . . . . 10 ⊢ ((𝐺‘𝑋) = (𝐾‘𝑋) → (((𝐺‘𝑋) ∈ ran 𝐺 ∧ (𝐾‘𝑋) ∈ ran 𝐾) ↔ ((𝐺‘𝑋) ∈ ran 𝐺 ∧ (𝐺‘𝑋) ∈ ran 𝐾)))
27		elin 4019	. . . . . . . . . . 11 ⊢ ((𝐺‘𝑋) ∈ (ran 𝐺 ∩ ran 𝐾) ↔ ((𝐺‘𝑋) ∈ ran 𝐺 ∧ (𝐺‘𝑋) ∈ ran 𝐾))
28	27	biimpri 220	. . . . . . . . . 10 ⊢ (((𝐺‘𝑋) ∈ ran 𝐺 ∧ (𝐺‘𝑋) ∈ ran 𝐾) → (𝐺‘𝑋) ∈ (ran 𝐺 ∩ ran 𝐾))
29	26, 28	syl6bi 245	. . . . . . . . 9 ⊢ ((𝐺‘𝑋) = (𝐾‘𝑋) → (((𝐺‘𝑋) ∈ ran 𝐺 ∧ (𝐾‘𝑋) ∈ ran 𝐾) → (𝐺‘𝑋) ∈ (ran 𝐺 ∩ ran 𝐾)))
30	23, 29	sylan9 503	. . . . . . . 8 ⊢ ((𝑋 ∈ (𝐴 ∩ 𝐵) ∧ (𝐺‘𝑋) = (𝐾‘𝑋)) → ((𝐺 Fn 𝐴 ∧ 𝐾 Fn 𝐵) → (𝐺‘𝑋) ∈ (ran 𝐺 ∩ ran 𝐾)))
31		fveq2 6446	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝐺‘𝑋) → (𝐹‘𝑥) = (𝐹‘(𝐺‘𝑋)))
32		fveq2 6446	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝐺‘𝑋) → (𝐻‘𝑥) = (𝐻‘(𝐺‘𝑋)))
33	31, 32	eqeq12d 2793	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐺‘𝑋) → ((𝐹‘𝑥) = (𝐻‘𝑥) ↔ (𝐹‘(𝐺‘𝑋)) = (𝐻‘(𝐺‘𝑋))))
34	33	rspcva 3509	. . . . . . . . . 10 ⊢ (((𝐺‘𝑋) ∈ (ran 𝐺 ∩ ran 𝐾) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹‘𝑥) = (𝐻‘𝑥)) → (𝐹‘(𝐺‘𝑋)) = (𝐻‘(𝐺‘𝑋)))
35	34	eqcomd 2784	. . . . . . . . 9 ⊢ (((𝐺‘𝑋) ∈ (ran 𝐺 ∩ ran 𝐾) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹‘𝑥) = (𝐻‘𝑥)) → (𝐻‘(𝐺‘𝑋)) = (𝐹‘(𝐺‘𝑋)))
36	35	ex 403	. . . . . . . 8 ⊢ ((𝐺‘𝑋) ∈ (ran 𝐺 ∩ ran 𝐾) → (∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹‘𝑥) = (𝐻‘𝑥) → (𝐻‘(𝐺‘𝑋)) = (𝐹‘(𝐺‘𝑋))))
37	30, 36	syl6 35	. . . . . . 7 ⊢ ((𝑋 ∈ (𝐴 ∩ 𝐵) ∧ (𝐺‘𝑋) = (𝐾‘𝑋)) → ((𝐺 Fn 𝐴 ∧ 𝐾 Fn 𝐵) → (∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹‘𝑥) = (𝐻‘𝑥) → (𝐻‘(𝐺‘𝑋)) = (𝐹‘(𝐺‘𝑋)))))
38	37	com23 86	. . . . . 6 ⊢ ((𝑋 ∈ (𝐴 ∩ 𝐵) ∧ (𝐺‘𝑋) = (𝐾‘𝑋)) → (∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹‘𝑥) = (𝐻‘𝑥) → ((𝐺 Fn 𝐴 ∧ 𝐾 Fn 𝐵) → (𝐻‘(𝐺‘𝑋)) = (𝐹‘(𝐺‘𝑋)))))
39	38	3impia 1106	. . . . 5 ⊢ ((𝑋 ∈ (𝐴 ∩ 𝐵) ∧ (𝐺‘𝑋) = (𝐾‘𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹‘𝑥) = (𝐻‘𝑥)) → ((𝐺 Fn 𝐴 ∧ 𝐾 Fn 𝐵) → (𝐻‘(𝐺‘𝑋)) = (𝐹‘(𝐺‘𝑋))))
40	39	impcom 398	. . . 4 ⊢ (((𝐺 Fn 𝐴 ∧ 𝐾 Fn 𝐵) ∧ (𝑋 ∈ (𝐴 ∩ 𝐵) ∧ (𝐺‘𝑋) = (𝐾‘𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹‘𝑥) = (𝐻‘𝑥))) → (𝐻‘(𝐺‘𝑋)) = (𝐹‘(𝐺‘𝑋)))
41	10, 14, 40	3eqtrrd 2819	. . 3 ⊢ (((𝐺 Fn 𝐴 ∧ 𝐾 Fn 𝐵) ∧ (𝑋 ∈ (𝐴 ∩ 𝐵) ∧ (𝐺‘𝑋) = (𝐾‘𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹‘𝑥) = (𝐻‘𝑥))) → (𝐹‘(𝐺‘𝑋)) = ((𝐻 ∘ 𝐾)‘𝑋))
42	5, 41	eqtrd 2814	. 2 ⊢ (((𝐺 Fn 𝐴 ∧ 𝐾 Fn 𝐵) ∧ (𝑋 ∈ (𝐴 ∩ 𝐵) ∧ (𝐺‘𝑋) = (𝐾‘𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹‘𝑥) = (𝐻‘𝑥))) → ((𝐹 ∘ 𝐺)‘𝑋) = ((𝐻 ∘ 𝐾)‘𝑋))
43	42	ex 403	1 ⊢ ((𝐺 Fn 𝐴 ∧ 𝐾 Fn 𝐵) → ((𝑋 ∈ (𝐴 ∩ 𝐵) ∧ (𝐺‘𝑋) = (𝐾‘𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹‘𝑥) = (𝐻‘𝑥)) → ((𝐹 ∘ 𝐺)‘𝑋) = ((𝐻 ∘ 𝐾)‘𝑋)))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1071   = wceq 1601   ∈ wcel 2107  ∀wral 3090   ∩ cin 3791  ran crn 5356   ∘ ccom 5359   Fn wfn 6130  ‘cfv 6135

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-fv 6143

This theorem is referenced by:  fvcosymgeq  18232