Theorem eqfnfv2d2 43297

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 3158	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))
4	1, 3	jca 521	. 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 521	. . 3 ⊢ (𝜑 → (𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵))
8		eqfnfv2 7061	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))))
9	7, 8	syl 18	. 2 ⊢ (𝜑 → (𝐹 = 𝐺 ↔ (𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))))
10	4, 9	mpbird 260	1 ⊢ (𝜑 → 𝐹 = 𝐺)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080   Fn wfn 6561  ‘cfv 6566

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-fv 6574

This theorem is used by: (None)