Theorem fvcofneq 6969

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 483	. . . 4 ⊢ ((𝐺 Fn 𝐴 ∧ 𝐾 Fn 𝐵) → 𝐺 Fn 𝐴)
2		elinel1 4129	. . . . 5 ⊢ (𝑋 ∈ (𝐴 ∩ 𝐵) → 𝑋 ∈ 𝐴)
3	2	3ad2ant1 1132	. . . 4 ⊢ ((𝑋 ∈ (𝐴 ∩ 𝐵) ∧ (𝐺‘𝑋) = (𝐾‘𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹‘𝑥) = (𝐻‘𝑥)) → 𝑋 ∈ 𝐴)
4		fvco2 6865	. . . 4 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋)))
5	1, 3, 4	syl2an 596	. . 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 485	. . . . 5 ⊢ ((𝐺 Fn 𝐴 ∧ 𝐾 Fn 𝐵) → 𝐾 Fn 𝐵)
7		elinel2 4130	. . . . . 6 ⊢ (𝑋 ∈ (𝐴 ∩ 𝐵) → 𝑋 ∈ 𝐵)
8	7	3ad2ant1 1132	. . . . 5 ⊢ ((𝑋 ∈ (𝐴 ∩ 𝐵) ∧ (𝐺‘𝑋) = (𝐾‘𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹‘𝑥) = (𝐻‘𝑥)) → 𝑋 ∈ 𝐵)
9		fvco2 6865	. . . . 5 ⊢ ((𝐾 Fn 𝐵 ∧ 𝑋 ∈ 𝐵) → ((𝐻 ∘ 𝐾)‘𝑋) = (𝐻‘(𝐾‘𝑋)))
10	6, 8, 9	syl2an 596	. . . 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 6774	. . . . . . 7 ⊢ ((𝐾‘𝑋) = (𝐺‘𝑋) → (𝐻‘(𝐾‘𝑋)) = (𝐻‘(𝐺‘𝑋)))
12	11	eqcoms 2746	. . . . . 6 ⊢ ((𝐺‘𝑋) = (𝐾‘𝑋) → (𝐻‘(𝐾‘𝑋)) = (𝐻‘(𝐺‘𝑋)))
13	12	3ad2ant2 1133	. . . . 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 482	. . . 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 6958	. . . . . . . . . . . 12 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐺‘𝑋) ∈ ran 𝐺)
17	15, 2, 16	syl2anr 597	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ (𝐴 ∩ 𝐵) ∧ 𝐺 Fn 𝐴) → (𝐺‘𝑋) ∈ ran 𝐺)
18	17	ex 413	. . . . . . . . . 10 ⊢ (𝑋 ∈ (𝐴 ∩ 𝐵) → (𝐺 Fn 𝐴 → (𝐺‘𝑋) ∈ ran 𝐺))
19		id 22	. . . . . . . . . . . 12 ⊢ (𝐾 Fn 𝐵 → 𝐾 Fn 𝐵)
20		fnfvelrn 6958	. . . . . . . . . . . 12 ⊢ ((𝐾 Fn 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝐾‘𝑋) ∈ ran 𝐾)
21	19, 7, 20	syl2anr 597	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ (𝐴 ∩ 𝐵) ∧ 𝐾 Fn 𝐵) → (𝐾‘𝑋) ∈ ran 𝐾)
22	21	ex 413	. . . . . . . . . 10 ⊢ (𝑋 ∈ (𝐴 ∩ 𝐵) → (𝐾 Fn 𝐵 → (𝐾‘𝑋) ∈ ran 𝐾))
23	18, 22	anim12d 609	. . . . . . . . 9 ⊢ (𝑋 ∈ (𝐴 ∩ 𝐵) → ((𝐺 Fn 𝐴 ∧ 𝐾 Fn 𝐵) → ((𝐺‘𝑋) ∈ ran 𝐺 ∧ (𝐾‘𝑋) ∈ ran 𝐾)))
24		eleq1 2826	. . . . . . . . . . . 12 ⊢ ((𝐾‘𝑋) = (𝐺‘𝑋) → ((𝐾‘𝑋) ∈ ran 𝐾 ↔ (𝐺‘𝑋) ∈ ran 𝐾))
25	24	eqcoms 2746	. . . . . . . . . . 11 ⊢ ((𝐺‘𝑋) = (𝐾‘𝑋) → ((𝐾‘𝑋) ∈ ran 𝐾 ↔ (𝐺‘𝑋) ∈ ran 𝐾))
26	25	anbi2d 629	. . . . . . . . . 10 ⊢ ((𝐺‘𝑋) = (𝐾‘𝑋) → (((𝐺‘𝑋) ∈ ran 𝐺 ∧ (𝐾‘𝑋) ∈ ran 𝐾) ↔ ((𝐺‘𝑋) ∈ ran 𝐺 ∧ (𝐺‘𝑋) ∈ ran 𝐾)))
27		elin 3903	. . . . . . . . . . 11 ⊢ ((𝐺‘𝑋) ∈ (ran 𝐺 ∩ ran 𝐾) ↔ ((𝐺‘𝑋) ∈ ran 𝐺 ∧ (𝐺‘𝑋) ∈ ran 𝐾))
28	27	biimpri 227	. . . . . . . . . 10 ⊢ (((𝐺‘𝑋) ∈ ran 𝐺 ∧ (𝐺‘𝑋) ∈ ran 𝐾) → (𝐺‘𝑋) ∈ (ran 𝐺 ∩ ran 𝐾))
29	26, 28	syl6bi 252	. . . . . . . . 9 ⊢ ((𝐺‘𝑋) = (𝐾‘𝑋) → (((𝐺‘𝑋) ∈ ran 𝐺 ∧ (𝐾‘𝑋) ∈ ran 𝐾) → (𝐺‘𝑋) ∈ (ran 𝐺 ∩ ran 𝐾)))
30	23, 29	sylan9 508	. . . . . . . 8 ⊢ ((𝑋 ∈ (𝐴 ∩ 𝐵) ∧ (𝐺‘𝑋) = (𝐾‘𝑋)) → ((𝐺 Fn 𝐴 ∧ 𝐾 Fn 𝐵) → (𝐺‘𝑋) ∈ (ran 𝐺 ∩ ran 𝐾)))
31		fveq2 6774	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝐺‘𝑋) → (𝐹‘𝑥) = (𝐹‘(𝐺‘𝑋)))
32		fveq2 6774	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝐺‘𝑋) → (𝐻‘𝑥) = (𝐻‘(𝐺‘𝑋)))
33	31, 32	eqeq12d 2754	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐺‘𝑋) → ((𝐹‘𝑥) = (𝐻‘𝑥) ↔ (𝐹‘(𝐺‘𝑋)) = (𝐻‘(𝐺‘𝑋))))
34	33	rspcva 3559	. . . . . . . . . 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 2744	. . . . . . . . 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 413	. . . . . . . 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 1116	. . . . 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 408	. . . 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 2783	. . 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 2778	. 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 413	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 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064   ∩ cin 3886  ran crn 5590   ∘ ccom 5593   Fn wfn 6428  ‘cfv 6433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-fv 6441

This theorem is referenced by:  fvcosymgeq  19037