Theorem ralrnmpt2 6775

Description: A restricted quantifier over an image set. (Contributed by Mario Carneiro, 1-Sep-2015.)

Hypotheses

Ref	Expression
rngop.1	⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
ralrnmpt2.2	⊢ (𝑧 = 𝐶 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
ralrnmpt2	⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → (∀𝑧 ∈ ran 𝐹𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓))

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

Allowed substitution hints:   𝜑(𝑧)   𝜓(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem ralrnmpt2

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rngop.1	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
2	1	rnmpt2 6770	. . . 4 ⊢ ran 𝐹 = {𝑤 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑤 = 𝐶}
3	2	raleqi 3142	. . 3 ⊢ (∀𝑧 ∈ ran 𝐹𝜑 ↔ ∀𝑧 ∈ {𝑤 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑤 = 𝐶}𝜑)
4		eqeq1 2626	. . . . 5 ⊢ (𝑤 = 𝑧 → (𝑤 = 𝐶 ↔ 𝑧 = 𝐶))
5	4	2rexbidv 3057	. . . 4 ⊢ (𝑤 = 𝑧 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑤 = 𝐶 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶))
6	5	ralab 3367	. . 3 ⊢ (∀𝑧 ∈ {𝑤 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑤 = 𝐶}𝜑 ↔ ∀𝑧(∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑))
7		ralcom4 3224	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑) ↔ ∀𝑧∀𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑))
8		r19.23v 3023	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑))
9	8	albii 1747	. . . 4 ⊢ (∀𝑧∀𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑) ↔ ∀𝑧(∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑))
10	7, 9	bitr2i 265	. . 3 ⊢ (∀𝑧(∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑) ↔ ∀𝑥 ∈ 𝐴 ∀𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑))
11	3, 6, 10	3bitri 286	. 2 ⊢ (∀𝑧 ∈ ran 𝐹𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑))
12		ralcom4 3224	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧(𝑧 = 𝐶 → 𝜑) ↔ ∀𝑧∀𝑦 ∈ 𝐵 (𝑧 = 𝐶 → 𝜑))
13		r19.23v 3023	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝐵 (𝑧 = 𝐶 → 𝜑) ↔ (∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑))
14	13	albii 1747	. . . . . 6 ⊢ (∀𝑧∀𝑦 ∈ 𝐵 (𝑧 = 𝐶 → 𝜑) ↔ ∀𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑))
15	12, 14	bitri 264	. . . . 5 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧(𝑧 = 𝐶 → 𝜑) ↔ ∀𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑))
16		nfv 1843	. . . . . . . 8 ⊢ Ⅎ𝑧𝜓
17		ralrnmpt2.2	. . . . . . . 8 ⊢ (𝑧 = 𝐶 → (𝜑 ↔ 𝜓))
18	16, 17	ceqsalg 3230	. . . . . . 7 ⊢ (𝐶 ∈ 𝑉 → (∀𝑧(𝑧 = 𝐶 → 𝜑) ↔ 𝜓))
19	18	ralimi 2952	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → ∀𝑦 ∈ 𝐵 (∀𝑧(𝑧 = 𝐶 → 𝜑) ↔ 𝜓))
20		ralbi 3068	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐵 (∀𝑧(𝑧 = 𝐶 → 𝜑) ↔ 𝜓) → (∀𝑦 ∈ 𝐵 ∀𝑧(𝑧 = 𝐶 → 𝜑) ↔ ∀𝑦 ∈ 𝐵 𝜓))
21	19, 20	syl 17	. . . . 5 ⊢ (∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → (∀𝑦 ∈ 𝐵 ∀𝑧(𝑧 = 𝐶 → 𝜑) ↔ ∀𝑦 ∈ 𝐵 𝜓))
22	15, 21	syl5bbr 274	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → (∀𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑) ↔ ∀𝑦 ∈ 𝐵 𝜓))
23	22	ralimi 2952	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 (∀𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑) ↔ ∀𝑦 ∈ 𝐵 𝜓))
24		ralbi 3068	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (∀𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑) ↔ ∀𝑦 ∈ 𝐵 𝜓) → (∀𝑥 ∈ 𝐴 ∀𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓))
25	23, 24	syl 17	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → (∀𝑥 ∈ 𝐴 ∀𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓))
26	11, 25	syl5bb 272	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → (∀𝑧 ∈ ran 𝐹𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓))

Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1481   = wceq 1483   ∈ wcel 1990  {cab 2608  ∀wral 2912  ∃wrex 2913  ran crn 5115   ↦ cmpt2 6652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-cnv 5122  df-dm 5124  df-rn 5125  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  rexrnmpt2  6776  efgval2  18137  txcnp  21423  txcnmpt  21427  txflf  21810