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Theorem eqfnfv2d2 41960
Description: Equality of functions is determined by their values, a deduction version. (Contributed by metakunt, 28-May-2024.)
Hypotheses
Ref Expression
eqfnfv2d2.1 (𝜑𝐹 Fn 𝐴)
eqfnfv2d2.2 (𝜑𝐺 Fn 𝐵)
eqfnfv2d2.3 (𝜑𝐴 = 𝐵)
eqfnfv2d2.4 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐺𝑥))
Assertion
Ref Expression
eqfnfv2d2 (𝜑𝐹 = 𝐺)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝐺   𝜑,𝑥
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem eqfnfv2d2
StepHypRef Expression
1 eqfnfv2d2.3 . . 3 (𝜑𝐴 = 𝐵)
2 eqfnfv2d2.4 . . . 4 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐺𝑥))
32ralrimiva 3145 . . 3 (𝜑 → ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥))
41, 3jca 511 . 2 (𝜑 → (𝐴 = 𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))
5 eqfnfv2d2.1 . . . 4 (𝜑𝐹 Fn 𝐴)
6 eqfnfv2d2.2 . . . 4 (𝜑𝐺 Fn 𝐵)
75, 6jca 511 . . 3 (𝜑 → (𝐹 Fn 𝐴𝐺 Fn 𝐵))
8 eqfnfv2 7050 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥))))
97, 8syl 17 . 2 (𝜑 → (𝐹 = 𝐺 ↔ (𝐴 = 𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥))))
104, 9mpbird 257 1 (𝜑𝐹 = 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3060   Fn wfn 6554  cfv 6559
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5294  ax-nul 5304  ax-pr 5430
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-br 5142  df-opab 5204  df-mpt 5224  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-iota 6512  df-fun 6561  df-fn 6562  df-fv 6567
This theorem is referenced by:  metakunt25  42208
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