Theorem ralrnmpt3 45168

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 3264	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉)
4		eqid 2740	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
5		ralrnmpt3.3	. . 3 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
6	4, 5	ralrnmptw 7128	. 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 206   ∧ wa 395   = wceq 1537  Ⅎwnf 1781   ∈ wcel 2108  ∀wral 3067   ↦ cmpt 5249  ran crn 5701

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-fv 6581

This theorem is referenced by:  liminflelimsuplem  45696