Theorem imaeqsalv 33594

Description: Substitute a function value into a universal quantifier over an image. (Contributed by Scott Fenton, 27-Sep-2024.)

Hypothesis

Ref	Expression
imaeqsex.1	⊢ (𝑥 = (𝐹‘𝑦) → (𝜑 ↔ 𝜓))

Assertion

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

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

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

Proof of Theorem imaeqsalv

Step	Hyp	Ref	Expression
1		imaeqsex.1	. . . . 5 ⊢ (𝑥 = (𝐹‘𝑦) → (𝜑 ↔ 𝜓))
2	1	notbid 317	. . . 4 ⊢ (𝑥 = (𝐹‘𝑦) → (¬ 𝜑 ↔ ¬ 𝜓))
3	2	imaeqsexv 33593	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∃𝑥 ∈ (𝐹 “ 𝐵) ¬ 𝜑 ↔ ∃𝑦 ∈ 𝐵 ¬ 𝜓))
4	3	notbid 317	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (¬ ∃𝑥 ∈ (𝐹 “ 𝐵) ¬ 𝜑 ↔ ¬ ∃𝑦 ∈ 𝐵 ¬ 𝜓))
5		dfral2 3164	. 2 ⊢ (∀𝑥 ∈ (𝐹 “ 𝐵)𝜑 ↔ ¬ ∃𝑥 ∈ (𝐹 “ 𝐵) ¬ 𝜑)
6		dfral2 3164	. 2 ⊢ (∀𝑦 ∈ 𝐵 𝜓 ↔ ¬ ∃𝑦 ∈ 𝐵 ¬ 𝜓)
7	4, 5, 6	3bitr4g 313	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∀𝑥 ∈ (𝐹 “ 𝐵)𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539  ∀wral 3063  ∃wrex 3064   ⊆ wss 3883   “ cima 5583   Fn wfn 6413  ‘cfv 6418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-fv 6426

This theorem is referenced by: (None)