Theorem imaeqsalv 7353

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 318	. . . 4 ⊢ (𝑥 = (𝐹‘𝑦) → (¬ 𝜑 ↔ ¬ 𝜓))
3	2	imaeqsexv 7352	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∃𝑥 ∈ (𝐹 “ 𝐵) ¬ 𝜑 ↔ ∃𝑦 ∈ 𝐵 ¬ 𝜓))
4	3	notbid 318	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (¬ ∃𝑥 ∈ (𝐹 “ 𝐵) ¬ 𝜑 ↔ ¬ ∃𝑦 ∈ 𝐵 ¬ 𝜓))
5		dfral2 3091	. 2 ⊢ (∀𝑥 ∈ (𝐹 “ 𝐵)𝜑 ↔ ¬ ∃𝑥 ∈ (𝐹 “ 𝐵) ¬ 𝜑)
6		dfral2 3091	. 2 ⊢ (∀𝑦 ∈ 𝐵 𝜓 ↔ ¬ ∃𝑦 ∈ 𝐵 ¬ 𝜓)
7	4, 5, 6	3bitr4g 314	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∀𝑥 ∈ (𝐹 “ 𝐵)𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533  ∀wral 3053  ∃wrex 3062   ⊆ wss 3940   “ cima 5669   Fn wfn 6528  ‘cfv 6533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-fv 6541

This theorem is referenced by:  negsunif  27882  negsbdaylem  27883