Theorem eqfnfv2d2 41931

Description: Equality of functions is determined by their values, a deduction version. (Contributed by metakunt, 28-May-2024.)

Hypotheses

Ref	Expression
eqfnfv2d2.1	⊢ (𝜑 → 𝐹 Fn 𝐴)
eqfnfv2d2.2	⊢ (𝜑 → 𝐺 Fn 𝐵)
eqfnfv2d2.3	⊢ (𝜑 → 𝐴 = 𝐵)
eqfnfv2d2.4	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐺‘𝑥))

Assertion

Ref	Expression
eqfnfv2d2	⊢ (𝜑 → 𝐹 = 𝐺)

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

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem eqfnfv2d2

Step	Hyp	Ref	Expression
1		eqfnfv2d2.3	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
2		eqfnfv2d2.4	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐺‘𝑥))
3	2	ralrimiva 3152	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))
4	1, 3	jca 511	. 2 ⊢ (𝜑 → (𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)))
5		eqfnfv2d2.1	. . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐴)
6		eqfnfv2d2.2	. . . 4 ⊢ (𝜑 → 𝐺 Fn 𝐵)
7	5, 6	jca 511	. . 3 ⊢ (𝜑 → (𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵))
8		eqfnfv2 7060	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))))
9	7, 8	syl 17	. 2 ⊢ (𝜑 → (𝐹 = 𝐺 ↔ (𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))))
10	4, 9	mpbird 257	1 ⊢ (𝜑 → 𝐹 = 𝐺)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067   Fn wfn 6563  ‘cfv 6568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5701  df-rel 5702  df-cnv 5703  df-co 5704  df-dm 5705  df-rn 5706  df-res 5707  df-ima 5708  df-iota 6520  df-fun 6570  df-fn 6571  df-fv 6576

This theorem is referenced by:  metakunt25  42179