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Theorem eqfnfv3 7047
Description: Derive equality of functions from equality of their values. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
eqfnfv3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐵𝐴 ∧ ∀𝑥𝐴 (𝑥𝐵 ∧ (𝐹𝑥) = (𝐺𝑥)))))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝐺   𝑥,𝐵

Proof of Theorem eqfnfv3
StepHypRef Expression
1 eqfnfv2 7046 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥))))
2 eqss 3997 . . . . 5 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
32biancomi 461 . . . 4 (𝐴 = 𝐵 ↔ (𝐵𝐴𝐴𝐵))
43anbi1i 622 . . 3 ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)) ↔ ((𝐵𝐴𝐴𝐵) ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))
5 anass 467 . . 3 (((𝐵𝐴𝐴𝐵) ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)) ↔ (𝐵𝐴 ∧ (𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥))))
6 dfss3 3970 . . . . . 6 (𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
76anbi1i 622 . . . . 5 ((𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)) ↔ (∀𝑥𝐴 𝑥𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))
8 r19.26 3108 . . . . 5 (∀𝑥𝐴 (𝑥𝐵 ∧ (𝐹𝑥) = (𝐺𝑥)) ↔ (∀𝑥𝐴 𝑥𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))
97, 8bitr4i 277 . . . 4 ((𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)) ↔ ∀𝑥𝐴 (𝑥𝐵 ∧ (𝐹𝑥) = (𝐺𝑥)))
109anbi2i 621 . . 3 ((𝐵𝐴 ∧ (𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥))) ↔ (𝐵𝐴 ∧ ∀𝑥𝐴 (𝑥𝐵 ∧ (𝐹𝑥) = (𝐺𝑥))))
114, 5, 103bitri 296 . 2 ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)) ↔ (𝐵𝐴 ∧ ∀𝑥𝐴 (𝑥𝐵 ∧ (𝐹𝑥) = (𝐺𝑥))))
121, 11bitrdi 286 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐵𝐴 ∧ ∀𝑥𝐴 (𝑥𝐵 ∧ (𝐹𝑥) = (𝐺𝑥)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wral 3058  wss 3949   Fn wfn 6548  cfv 6553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-fv 6561
This theorem is referenced by: (None)
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