Theorem ralrnmpt3 45717

Description: A restricted quantifier over an image set. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypotheses

Ref	Expression
ralrnmpt3.1	⊢ Ⅎ𝑥𝜑
ralrnmpt3.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
ralrnmpt3.3	⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
ralrnmpt3	⊢ (𝜑 → (∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑦,𝐵   𝜒,𝑦   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)   𝜒(𝑥)   𝐵(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem ralrnmpt3

Step	Hyp	Ref	Expression
1		ralrnmpt3.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		ralrnmpt3.2	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
3	1, 2	ralrimia 3240	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉)
4		eqid 2741	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
5		ralrnmpt3.3	. . 3 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
6	4, 5	ralrnmptw 7039	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒))
7	3, 6	syl 17	1 ⊢ (𝜑 → (∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   = wceq 1548  Ⅎwnf 1791   ∈ wcel 2121  ∀wral 3055   ↦ cmpt 5156  ran crn 5622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-mpt 5157  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:  liminflelimsuplem  46232