Theorem ralimaOLD 7238

Description: Obsolete version of ralima 7235 as of 14-Aug-2025. (Contributed by Stefan O'Rear, 21-Jan-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
reximaOLD.x	⊢ (𝑥 = (𝐹‘𝑦) → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
ralimaOLD	⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∀𝑥 ∈ (𝐹 “ 𝐵)𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜓))

Distinct variable groups:   𝜑,𝑦   𝜓,𝑥   𝑥,𝐹,𝑦   𝑥,𝐵,𝑦   𝑥,𝐴,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem ralimaOLD

Step	Hyp	Ref	Expression
1		fvexd 6896	. 2 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝐹‘𝑦) ∈ V)
2		fvelimab 6953	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑥 ∈ (𝐹 “ 𝐵) ↔ ∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑥))
3		eqcom 2769	. . . 4 ⊢ ((𝐹‘𝑦) = 𝑥 ↔ 𝑥 = (𝐹‘𝑦))
4	3	rexbii 3111	. . 3 ⊢ (∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑥 ↔ ∃𝑦 ∈ 𝐵 𝑥 = (𝐹‘𝑦))
5	2, 4	bitrdi 290	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑥 ∈ (𝐹 “ 𝐵) ↔ ∃𝑦 ∈ 𝐵 𝑥 = (𝐹‘𝑦)))
6		reximaOLD.x	. . 3 ⊢ (𝑥 = (𝐹‘𝑦) → (𝜑 ↔ 𝜓))
7	6	adantl 486	. 2 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 = (𝐹‘𝑦)) → (𝜑 ↔ 𝜓))
8	1, 5, 7	ralxfr2d 5380	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∀𝑥 ∈ (𝐹 “ 𝐵)𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1569   ∈ wcel 2142  ∀wral 3078  ∃wrex 3088  Vcvv 3454   ⊆ wss 3904   “ cima 5663   Fn wfn 6531  ‘cfv 6536

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-fv 6544

This theorem is used by: (None)