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Theorem eqfnfv2d2 40468
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 3144 . . 3 (𝜑 → ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥))
41, 3jca 513 . 2 (𝜑 → (𝐴 = 𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))
5 eqfnfv2d2.1 . . . 4 (𝜑𝐹 Fn 𝐴)
6 eqfnfv2d2.2 . . . 4 (𝜑𝐺 Fn 𝐵)
75, 6jca 513 . . 3 (𝜑 → (𝐹 Fn 𝐴𝐺 Fn 𝐵))
8 eqfnfv2 6988 . . 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 205  wa 397   = wceq 1542  wcel 2107  wral 3065   Fn wfn 6496  cfv 6501
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-fv 6509
This theorem is referenced by:  metakunt25  40630
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