Theorem eqfnfv2d2 42147

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 3125	. . 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 6974	. . 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 1541   ∈ wcel 2113  ∀wral 3048   Fn wfn 6484  ‘cfv 6489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-fv 6497

This theorem is referenced by: (None)