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Theorem eqfnfv3 6623
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 6622 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥))))
2 eqss 3869 . . . . 5 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
32biancomi 455 . . . 4 (𝐴 = 𝐵 ↔ (𝐵𝐴𝐴𝐵))
43anbi1i 614 . . 3 ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)) ↔ ((𝐵𝐴𝐴𝐵) ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))
5 anass 461 . . 3 (((𝐵𝐴𝐴𝐵) ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)) ↔ (𝐵𝐴 ∧ (𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥))))
6 dfss3 3843 . . . . . 6 (𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
76anbi1i 614 . . . . 5 ((𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)) ↔ (∀𝑥𝐴 𝑥𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))
8 r19.26 3114 . . . . 5 (∀𝑥𝐴 (𝑥𝐵 ∧ (𝐹𝑥) = (𝐺𝑥)) ↔ (∀𝑥𝐴 𝑥𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))
97, 8bitr4i 270 . . . 4 ((𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)) ↔ ∀𝑥𝐴 (𝑥𝐵 ∧ (𝐹𝑥) = (𝐺𝑥)))
109anbi2i 613 . . 3 ((𝐵𝐴 ∧ (𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥))) ↔ (𝐵𝐴 ∧ ∀𝑥𝐴 (𝑥𝐵 ∧ (𝐹𝑥) = (𝐺𝑥))))
114, 5, 103bitri 289 . 2 ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)) ↔ (𝐵𝐴 ∧ ∀𝑥𝐴 (𝑥𝐵 ∧ (𝐹𝑥) = (𝐺𝑥))))
121, 11syl6bb 279 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐵𝐴 ∧ ∀𝑥𝐴 (𝑥𝐵 ∧ (𝐹𝑥) = (𝐺𝑥)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387   = wceq 1507  wcel 2048  wral 3082  wss 3825   Fn wfn 6177  cfv 6182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5305  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-iota 6146  df-fun 6184  df-fn 6185  df-fv 6190
This theorem is referenced by: (None)
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