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Theorem fvcofneq 6275
 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 3758 . . . . . 6 (𝑋 ∈ (𝐴𝐵) ↔ (𝑋𝐴𝑋𝐵))
3 simpl 472 . . . . . 6 ((𝑋𝐴𝑋𝐵) → 𝑋𝐴)
42, 3sylbi 206 . . . . 5 (𝑋 ∈ (𝐴𝐵) → 𝑋𝐴)
543ad2ant1 1075 . . . 4 ((𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥)) → 𝑋𝐴)
6 fvco2 6183 . . . 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 206 . . . . . 6 (𝑋 ∈ (𝐴𝐵) → 𝑋𝐵)
11103ad2ant1 1075 . . . . 5 ((𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥)) → 𝑋𝐵)
12 fvco2 6183 . . . . 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 6103 . . . . . . 7 ((𝐾𝑋) = (𝐺𝑋) → (𝐻‘(𝐾𝑋)) = (𝐻‘(𝐺𝑋)))
1514eqcoms 2618 . . . . . 6 ((𝐺𝑋) = (𝐾𝑋) → (𝐻‘(𝐾𝑋)) = (𝐻‘(𝐺𝑋)))
16153ad2ant2 1076 . . . . 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 6264 . . . . . . . . . . . 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 6264 . . . . . . . . . . . 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 584 . . . . . . . . 9 (𝑋 ∈ (𝐴𝐵) → ((𝐺 Fn 𝐴𝐾 Fn 𝐵) → ((𝐺𝑋) ∈ ran 𝐺 ∧ (𝐾𝑋) ∈ ran 𝐾)))
27 eleq1 2676 . . . . . . . . . . . 12 ((𝐾𝑋) = (𝐺𝑋) → ((𝐾𝑋) ∈ ran 𝐾 ↔ (𝐺𝑋) ∈ ran 𝐾))
2827eqcoms 2618 . . . . . . . . . . 11 ((𝐺𝑋) = (𝐾𝑋) → ((𝐾𝑋) ∈ ran 𝐾 ↔ (𝐺𝑋) ∈ ran 𝐾))
2928anbi2d 736 . . . . . . . . . 10 ((𝐺𝑋) = (𝐾𝑋) → (((𝐺𝑋) ∈ ran 𝐺 ∧ (𝐾𝑋) ∈ ran 𝐾) ↔ ((𝐺𝑋) ∈ ran 𝐺 ∧ (𝐺𝑋) ∈ ran 𝐾)))
30 elin 3758 . . . . . . . . . . 11 ((𝐺𝑋) ∈ (ran 𝐺 ∩ ran 𝐾) ↔ ((𝐺𝑋) ∈ ran 𝐺 ∧ (𝐺𝑋) ∈ ran 𝐾))
3130biimpri 217 . . . . . . . . . 10 (((𝐺𝑋) ∈ ran 𝐺 ∧ (𝐺𝑋) ∈ ran 𝐾) → (𝐺𝑋) ∈ (ran 𝐺 ∩ ran 𝐾))
3229, 31syl6bi 242 . . . . . . . . 9 ((𝐺𝑋) = (𝐾𝑋) → (((𝐺𝑋) ∈ ran 𝐺 ∧ (𝐾𝑋) ∈ ran 𝐾) → (𝐺𝑋) ∈ (ran 𝐺 ∩ ran 𝐾)))
3326, 32sylan9 687 . . . . . . . 8 ((𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋)) → ((𝐺 Fn 𝐴𝐾 Fn 𝐵) → (𝐺𝑋) ∈ (ran 𝐺 ∩ ran 𝐾)))
34 fveq2 6103 . . . . . . . . . . . 12 (𝑥 = (𝐺𝑋) → (𝐹𝑥) = (𝐹‘(𝐺𝑋)))
35 fveq2 6103 . . . . . . . . . . . 12 (𝑥 = (𝐺𝑋) → (𝐻𝑥) = (𝐻‘(𝐺𝑋)))
3634, 35eqeq12d 2625 . . . . . . . . . . 11 (𝑥 = (𝐺𝑋) → ((𝐹𝑥) = (𝐻𝑥) ↔ (𝐹‘(𝐺𝑋)) = (𝐻‘(𝐺𝑋))))
3736rspcva 3280 . . . . . . . . . 10 (((𝐺𝑋) ∈ (ran 𝐺 ∩ ran 𝐾) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥)) → (𝐹‘(𝐺𝑋)) = (𝐻‘(𝐺𝑋)))
3837eqcomd 2616 . . . . . . . . 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 34 . . . . . . 7 ((𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋)) → ((𝐺 Fn 𝐴𝐾 Fn 𝐵) → (∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥) → (𝐻‘(𝐺𝑋)) = (𝐹‘(𝐺𝑋)))))
4140com23 84 . . . . . 6 ((𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋)) → (∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥) → ((𝐺 Fn 𝐴𝐾 Fn 𝐵) → (𝐻‘(𝐺𝑋)) = (𝐹‘(𝐺𝑋)))))
42413impia 1253 . . . . 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 2649 . . 3 (((𝐺 Fn 𝐴𝐾 Fn 𝐵) ∧ (𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥))) → (𝐹‘(𝐺𝑋)) = ((𝐻𝐾)‘𝑋))
457, 44eqtrd 2644 . 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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ∩ cin 3539  ran crn 5039   ∘ ccom 5042   Fn wfn 5799  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812 This theorem is referenced by:  fvcosymgeq  17672
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