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Theorem fvcofneq 6407
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 𝐴𝐾 Fn 𝐵) → ((𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥)) → ((𝐹𝐺)‘𝑋) = ((𝐻𝐾)‘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
StepHypRef Expression
1 simpl 472 . . . 4 ((𝐺 Fn 𝐴𝐾 Fn 𝐵) → 𝐺 Fn 𝐴)
2 elin 3829 . . . . . 6 (𝑋 ∈ (𝐴𝐵) ↔ (𝑋𝐴𝑋𝐵))
3 simpl 472 . . . . . 6 ((𝑋𝐴𝑋𝐵) → 𝑋𝐴)
42, 3sylbi 207 . . . . 5 (𝑋 ∈ (𝐴𝐵) → 𝑋𝐴)
543ad2ant1 1102 . . . 4 ((𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥)) → 𝑋𝐴)
6 fvco2 6312 . . . 4 ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))
71, 5, 6syl2an 493 . . 3 (((𝐺 Fn 𝐴𝐾 Fn 𝐵) ∧ (𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥))) → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))
8 simpr 476 . . . . 5 ((𝐺 Fn 𝐴𝐾 Fn 𝐵) → 𝐾 Fn 𝐵)
9 simpr 476 . . . . . . 7 ((𝑋𝐴𝑋𝐵) → 𝑋𝐵)
102, 9sylbi 207 . . . . . 6 (𝑋 ∈ (𝐴𝐵) → 𝑋𝐵)
11103ad2ant1 1102 . . . . 5 ((𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥)) → 𝑋𝐵)
12 fvco2 6312 . . . . 5 ((𝐾 Fn 𝐵𝑋𝐵) → ((𝐻𝐾)‘𝑋) = (𝐻‘(𝐾𝑋)))
138, 11, 12syl2an 493 . . . 4 (((𝐺 Fn 𝐴𝐾 Fn 𝐵) ∧ (𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥))) → ((𝐻𝐾)‘𝑋) = (𝐻‘(𝐾𝑋)))
14 fveq2 6229 . . . . . . 7 ((𝐾𝑋) = (𝐺𝑋) → (𝐻‘(𝐾𝑋)) = (𝐻‘(𝐺𝑋)))
1514eqcoms 2659 . . . . . 6 ((𝐺𝑋) = (𝐾𝑋) → (𝐻‘(𝐾𝑋)) = (𝐻‘(𝐺𝑋)))
16153ad2ant2 1103 . . . . 5 ((𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥)) → (𝐻‘(𝐾𝑋)) = (𝐻‘(𝐺𝑋)))
1716adantl 481 . . . 4 (((𝐺 Fn 𝐴𝐾 Fn 𝐵) ∧ (𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥))) → (𝐻‘(𝐾𝑋)) = (𝐻‘(𝐺𝑋)))
18 id 22 . . . . . . . . . . . 12 (𝐺 Fn 𝐴𝐺 Fn 𝐴)
19 fnfvelrn 6396 . . . . . . . . . . . 12 ((𝐺 Fn 𝐴𝑋𝐴) → (𝐺𝑋) ∈ ran 𝐺)
2018, 4, 19syl2anr 494 . . . . . . . . . . 11 ((𝑋 ∈ (𝐴𝐵) ∧ 𝐺 Fn 𝐴) → (𝐺𝑋) ∈ ran 𝐺)
2120ex 449 . . . . . . . . . 10 (𝑋 ∈ (𝐴𝐵) → (𝐺 Fn 𝐴 → (𝐺𝑋) ∈ ran 𝐺))
22 id 22 . . . . . . . . . . . 12 (𝐾 Fn 𝐵𝐾 Fn 𝐵)
23 fnfvelrn 6396 . . . . . . . . . . . 12 ((𝐾 Fn 𝐵𝑋𝐵) → (𝐾𝑋) ∈ ran 𝐾)
2422, 10, 23syl2anr 494 . . . . . . . . . . 11 ((𝑋 ∈ (𝐴𝐵) ∧ 𝐾 Fn 𝐵) → (𝐾𝑋) ∈ ran 𝐾)
2524ex 449 . . . . . . . . . 10 (𝑋 ∈ (𝐴𝐵) → (𝐾 Fn 𝐵 → (𝐾𝑋) ∈ ran 𝐾))
2621, 25anim12d 585 . . . . . . . . 9 (𝑋 ∈ (𝐴𝐵) → ((𝐺 Fn 𝐴𝐾 Fn 𝐵) → ((𝐺𝑋) ∈ ran 𝐺 ∧ (𝐾𝑋) ∈ ran 𝐾)))
27 eleq1 2718 . . . . . . . . . . . 12 ((𝐾𝑋) = (𝐺𝑋) → ((𝐾𝑋) ∈ ran 𝐾 ↔ (𝐺𝑋) ∈ ran 𝐾))
2827eqcoms 2659 . . . . . . . . . . 11 ((𝐺𝑋) = (𝐾𝑋) → ((𝐾𝑋) ∈ ran 𝐾 ↔ (𝐺𝑋) ∈ ran 𝐾))
2928anbi2d 740 . . . . . . . . . 10 ((𝐺𝑋) = (𝐾𝑋) → (((𝐺𝑋) ∈ ran 𝐺 ∧ (𝐾𝑋) ∈ ran 𝐾) ↔ ((𝐺𝑋) ∈ ran 𝐺 ∧ (𝐺𝑋) ∈ ran 𝐾)))
30 elin 3829 . . . . . . . . . . 11 ((𝐺𝑋) ∈ (ran 𝐺 ∩ ran 𝐾) ↔ ((𝐺𝑋) ∈ ran 𝐺 ∧ (𝐺𝑋) ∈ ran 𝐾))
3130biimpri 218 . . . . . . . . . 10 (((𝐺𝑋) ∈ ran 𝐺 ∧ (𝐺𝑋) ∈ ran 𝐾) → (𝐺𝑋) ∈ (ran 𝐺 ∩ ran 𝐾))
3229, 31syl6bi 243 . . . . . . . . 9 ((𝐺𝑋) = (𝐾𝑋) → (((𝐺𝑋) ∈ ran 𝐺 ∧ (𝐾𝑋) ∈ ran 𝐾) → (𝐺𝑋) ∈ (ran 𝐺 ∩ ran 𝐾)))
3326, 32sylan9 690 . . . . . . . 8 ((𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋)) → ((𝐺 Fn 𝐴𝐾 Fn 𝐵) → (𝐺𝑋) ∈ (ran 𝐺 ∩ ran 𝐾)))
34 fveq2 6229 . . . . . . . . . . . 12 (𝑥 = (𝐺𝑋) → (𝐹𝑥) = (𝐹‘(𝐺𝑋)))
35 fveq2 6229 . . . . . . . . . . . 12 (𝑥 = (𝐺𝑋) → (𝐻𝑥) = (𝐻‘(𝐺𝑋)))
3634, 35eqeq12d 2666 . . . . . . . . . . 11 (𝑥 = (𝐺𝑋) → ((𝐹𝑥) = (𝐻𝑥) ↔ (𝐹‘(𝐺𝑋)) = (𝐻‘(𝐺𝑋))))
3736rspcva 3338 . . . . . . . . . 10 (((𝐺𝑋) ∈ (ran 𝐺 ∩ ran 𝐾) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥)) → (𝐹‘(𝐺𝑋)) = (𝐻‘(𝐺𝑋)))
3837eqcomd 2657 . . . . . . . . 9 (((𝐺𝑋) ∈ (ran 𝐺 ∩ ran 𝐾) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥)) → (𝐻‘(𝐺𝑋)) = (𝐹‘(𝐺𝑋)))
3938ex 449 . . . . . . . 8 ((𝐺𝑋) ∈ (ran 𝐺 ∩ ran 𝐾) → (∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥) → (𝐻‘(𝐺𝑋)) = (𝐹‘(𝐺𝑋))))
4033, 39syl6 35 . . . . . . 7 ((𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋)) → ((𝐺 Fn 𝐴𝐾 Fn 𝐵) → (∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥) → (𝐻‘(𝐺𝑋)) = (𝐹‘(𝐺𝑋)))))
4140com23 86 . . . . . 6 ((𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋)) → (∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥) → ((𝐺 Fn 𝐴𝐾 Fn 𝐵) → (𝐻‘(𝐺𝑋)) = (𝐹‘(𝐺𝑋)))))
42413impia 1280 . . . . 5 ((𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥)) → ((𝐺 Fn 𝐴𝐾 Fn 𝐵) → (𝐻‘(𝐺𝑋)) = (𝐹‘(𝐺𝑋))))
4342impcom 445 . . . 4 (((𝐺 Fn 𝐴𝐾 Fn 𝐵) ∧ (𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥))) → (𝐻‘(𝐺𝑋)) = (𝐹‘(𝐺𝑋)))
4413, 17, 433eqtrrd 2690 . . 3 (((𝐺 Fn 𝐴𝐾 Fn 𝐵) ∧ (𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥))) → (𝐹‘(𝐺𝑋)) = ((𝐻𝐾)‘𝑋))
457, 44eqtrd 2685 . 2 (((𝐺 Fn 𝐴𝐾 Fn 𝐵) ∧ (𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥))) → ((𝐹𝐺)‘𝑋) = ((𝐻𝐾)‘𝑋))
4645ex 449 1 ((𝐺 Fn 𝐴𝐾 Fn 𝐵) → ((𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥)) → ((𝐹𝐺)‘𝑋) = ((𝐻𝐾)‘𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  cin 3606  ran crn 5144  ccom 5147   Fn wfn 5921  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934
This theorem is referenced by:  fvcosymgeq  17895
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