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Theorem eqfnfv3 6474
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 6473 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥))))
2 eqss 3757 . . . . 5 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
3 ancom 465 . . . . 5 ((𝐴𝐵𝐵𝐴) ↔ (𝐵𝐴𝐴𝐵))
42, 3bitri 264 . . . 4 (𝐴 = 𝐵 ↔ (𝐵𝐴𝐴𝐵))
54anbi1i 733 . . 3 ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)) ↔ ((𝐵𝐴𝐴𝐵) ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))
6 anass 684 . . . 4 (((𝐵𝐴𝐴𝐵) ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)) ↔ (𝐵𝐴 ∧ (𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥))))
7 dfss3 3731 . . . . . . 7 (𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
87anbi1i 733 . . . . . 6 ((𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)) ↔ (∀𝑥𝐴 𝑥𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))
9 r19.26 3200 . . . . . 6 (∀𝑥𝐴 (𝑥𝐵 ∧ (𝐹𝑥) = (𝐺𝑥)) ↔ (∀𝑥𝐴 𝑥𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))
108, 9bitr4i 267 . . . . 5 ((𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)) ↔ ∀𝑥𝐴 (𝑥𝐵 ∧ (𝐹𝑥) = (𝐺𝑥)))
1110anbi2i 732 . . . 4 ((𝐵𝐴 ∧ (𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥))) ↔ (𝐵𝐴 ∧ ∀𝑥𝐴 (𝑥𝐵 ∧ (𝐹𝑥) = (𝐺𝑥))))
126, 11bitri 264 . . 3 (((𝐵𝐴𝐴𝐵) ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)) ↔ (𝐵𝐴 ∧ ∀𝑥𝐴 (𝑥𝐵 ∧ (𝐹𝑥) = (𝐺𝑥))))
135, 12bitri 264 . 2 ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)) ↔ (𝐵𝐴 ∧ ∀𝑥𝐴 (𝑥𝐵 ∧ (𝐹𝑥) = (𝐺𝑥))))
141, 13syl6bb 276 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐵𝐴 ∧ ∀𝑥𝐴 (𝑥𝐵 ∧ (𝐹𝑥) = (𝐺𝑥)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1630  wcel 2137  wral 3048  wss 3713   Fn wfn 6042  cfv 6047
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ral 3053  df-rex 3054  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-sn 4320  df-pr 4322  df-op 4326  df-uni 4587  df-br 4803  df-opab 4863  df-mpt 4880  df-id 5172  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-iota 6010  df-fun 6049  df-fn 6050  df-fv 6055
This theorem is referenced by: (None)
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