Theorem rexrnmpo 6034

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)
ralrnmpo.2	⊢ (𝑧 = 𝐶 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
rexrnmpo	⊢ (∀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 rexrnmpo

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	rnmpo 6029	. . . 4 ⊢ ran 𝐹 = {𝑤 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑤 = 𝐶}
3	2	rexeqi 2695	. . 3 ⊢ (∃𝑧 ∈ ran 𝐹𝜑 ↔ ∃𝑧 ∈ {𝑤 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑤 = 𝐶}𝜑)
4		eqeq1 2200	. . . . 5 ⊢ (𝑤 = 𝑧 → (𝑤 = 𝐶 ↔ 𝑧 = 𝐶))
5	4	2rexbidv 2519	. . . 4 ⊢ (𝑤 = 𝑧 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑤 = 𝐶 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶))
6	5	rexab 2922	. . 3 ⊢ (∃𝑧 ∈ {𝑤 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑤 = 𝐶}𝜑 ↔ ∃𝑧(∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑))
7		rexcom4 2783	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑) ↔ ∃𝑧∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑))
8		r19.41v 2650	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑))
9	8	exbii 1616	. . . 4 ⊢ (∃𝑧∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑) ↔ ∃𝑧(∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑))
10	7, 9	bitr2i 185	. . 3 ⊢ (∃𝑧(∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑) ↔ ∃𝑥 ∈ 𝐴 ∃𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑))
11	3, 6, 10	3bitri 206	. 2 ⊢ (∃𝑧 ∈ ran 𝐹𝜑 ↔ ∃𝑥 ∈ 𝐴 ∃𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑))
12		rexcom4 2783	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑧(𝑧 = 𝐶 ∧ 𝜑) ↔ ∃𝑧∃𝑦 ∈ 𝐵 (𝑧 = 𝐶 ∧ 𝜑))
13		r19.41v 2650	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝐵 (𝑧 = 𝐶 ∧ 𝜑) ↔ (∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑))
14	13	exbii 1616	. . . . . 6 ⊢ (∃𝑧∃𝑦 ∈ 𝐵 (𝑧 = 𝐶 ∧ 𝜑) ↔ ∃𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑))
15	12, 14	bitri 184	. . . . 5 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑧(𝑧 = 𝐶 ∧ 𝜑) ↔ ∃𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑))
16		ralrnmpo.2	. . . . . . . 8 ⊢ (𝑧 = 𝐶 → (𝜑 ↔ 𝜓))
17	16	ceqsexgv 2889	. . . . . . 7 ⊢ (𝐶 ∈ 𝑉 → (∃𝑧(𝑧 = 𝐶 ∧ 𝜑) ↔ 𝜓))
18	17	ralimi 2557	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → ∀𝑦 ∈ 𝐵 (∃𝑧(𝑧 = 𝐶 ∧ 𝜑) ↔ 𝜓))
19		rexbi 2627	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐵 (∃𝑧(𝑧 = 𝐶 ∧ 𝜑) ↔ 𝜓) → (∃𝑦 ∈ 𝐵 ∃𝑧(𝑧 = 𝐶 ∧ 𝜑) ↔ ∃𝑦 ∈ 𝐵 𝜓))
20	18, 19	syl 14	. . . . 5 ⊢ (∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → (∃𝑦 ∈ 𝐵 ∃𝑧(𝑧 = 𝐶 ∧ 𝜑) ↔ ∃𝑦 ∈ 𝐵 𝜓))
21	15, 20	bitr3id 194	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → (∃𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑) ↔ ∃𝑦 ∈ 𝐵 𝜓))
22	21	ralimi 2557	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 (∃𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑) ↔ ∃𝑦 ∈ 𝐵 𝜓))
23		rexbi 2627	. . 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))
24	22, 23	syl 14	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → (∃𝑥 ∈ 𝐴 ∃𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ∧ 𝜑) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓))
25	11, 24	bitrid 192	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → (∃𝑧 ∈ ran 𝐹𝜑 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364  ∃wex 1503   ∈ wcel 2164  {cab 2179  ∀wral 2472  ∃wrex 2473  ran crn 4660   ∈ cmpo 5920

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-cnv 4667  df-dm 4669  df-rn 4670  df-oprab 5922  df-mpo 5923

This theorem is referenced by:  eltx  14427  txrest  14444  txlm  14447