Theorem eqfnfv2 5754

Description: Equality of functions is determined by their values. Exercise 4 of [TakeutiZaring] p. 28. (Contributed by NM, 3-Aug-1994.) (Revised by Mario Carneiro, 31-Aug-2015.)

Assertion

Ref	Expression
eqfnfv2	⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝐺

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem eqfnfv2

Step	Hyp	Ref	Expression
1		dmeq 4937	. . . 4 ⊢ (𝐹 = 𝐺 → dom 𝐹 = dom 𝐺)
2		fndm 5436	. . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
3		fndm 5436	. . . . 5 ⊢ (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
4	2, 3	eqeqan12d 2247	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → (dom 𝐹 = dom 𝐺 ↔ 𝐴 = 𝐵))
5	1, 4	imbitrid 154	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → (𝐹 = 𝐺 → 𝐴 = 𝐵))
6	5	pm4.71rd 394	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐵 ∧ 𝐹 = 𝐺)))
7		fneq2 5426	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐺 Fn 𝐴 ↔ 𝐺 Fn 𝐵))
8	7	biimparc 299	. . . . 5 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐴 = 𝐵) → 𝐺 Fn 𝐴)
9		eqfnfv 5753	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)))
10	8, 9	sylan2 286	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐺 Fn 𝐵 ∧ 𝐴 = 𝐵)) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)))
11	10	anassrs 400	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ 𝐴 = 𝐵) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)))
12	11	pm5.32da 452	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → ((𝐴 = 𝐵 ∧ 𝐹 = 𝐺) ↔ (𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))))
13	6, 12	bitrd 188	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398  ∀wral 2511  dom cdm 4731   Fn wfn 5328  ‘cfv 5333

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341

This theorem is referenced by:  eqfnfv3  5755  eqfunfv  5758  eqfnov  6138  2ffzeq  10421  eqwrd  11203