Theorem rexima 7244

Description: Existential quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.) Reduce DV conditions. (Revised by Matthew House, 14-Aug-2025.)

Hypothesis

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

Assertion

Ref	Expression
rexima	⊢ ((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

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

Proof of Theorem rexima

Step	Hyp	Ref	Expression
1		ralima.x	. . . . 5 ⊢ (𝑥 = (𝐹‘𝑦) → (𝜑 ↔ 𝜓))
2	1	notbid 321	. . . 4 ⊢ (𝑥 = (𝐹‘𝑦) → (¬ 𝜑 ↔ ¬ 𝜓))
3	2	ralima 7243	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∀𝑥 ∈ (𝐹 “ 𝐵) ¬ 𝜑 ↔ ∀𝑦 ∈ 𝐵 ¬ 𝜓))
4	3	notbid 321	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (¬ ∀𝑥 ∈ (𝐹 “ 𝐵) ¬ 𝜑 ↔ ¬ ∀𝑦 ∈ 𝐵 ¬ 𝜓))
5		dfrex2 3089	. 2 ⊢ (∃𝑥 ∈ (𝐹 “ 𝐵)𝜑 ↔ ¬ ∀𝑥 ∈ (𝐹 “ 𝐵) ¬ 𝜑)
6		dfrex2 3089	. 2 ⊢ (∃𝑦 ∈ 𝐵 𝜓 ↔ ¬ ∀𝑦 ∈ 𝐵 ¬ 𝜓)
7	4, 5, 6	3bitr4g 317	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∃𝑥 ∈ (𝐹 “ 𝐵)𝜑 ↔ ∃𝑦 ∈ 𝐵 𝜓))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569  ∀wral 3076  ∃wrex 3086   ⊆ wss 3895   “ cima 5653   Fn wfn 6533  ‘cfv 6538

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-fv 6546

This theorem is used by:  supisolem  9469  ipodrsima  18720  lmflf  24329  caucfil  25609  dyadmbllem  25925  lhop1lem  26338  negsid  28432  negsunif  28446  nummin  35774  vonf1wev  35955  vonf1owevOLD  35957  mblfinlem1  38668  itg2gt0cn  38686