Theorem rexrnmpt 5701

Description: A restricted quantifier over an image set. (Contributed by Mario Carneiro, 20-Aug-2015.)

Hypotheses

Ref	Expression
ralrnmpt.1	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
ralrnmpt.2	⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
rexrnmpt	⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∃𝑦 ∈ ran 𝐹𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒))

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

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

Proof of Theorem rexrnmpt

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ralrnmpt.1	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
2	1	fnmpt 5380	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → 𝐹 Fn 𝐴)
3		dfsbcq 2987	. . . . 5 ⊢ (𝑤 = (𝐹‘𝑧) → ([𝑤 / 𝑦]𝜓 ↔ [(𝐹‘𝑧) / 𝑦]𝜓))
4	3	rexrn 5695	. . . 4 ⊢ (𝐹 Fn 𝐴 → (∃𝑤 ∈ ran 𝐹[𝑤 / 𝑦]𝜓 ↔ ∃𝑧 ∈ 𝐴 [(𝐹‘𝑧) / 𝑦]𝜓))
5	2, 4	syl 14	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∃𝑤 ∈ ran 𝐹[𝑤 / 𝑦]𝜓 ↔ ∃𝑧 ∈ 𝐴 [(𝐹‘𝑧) / 𝑦]𝜓))
6		nfv 1539	. . . . 5 ⊢ Ⅎ𝑤𝜓
7		nfsbc1v 3004	. . . . 5 ⊢ Ⅎ𝑦[𝑤 / 𝑦]𝜓
8		sbceq1a 2995	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝜓 ↔ [𝑤 / 𝑦]𝜓))
9	6, 7, 8	cbvrex 2723	. . . 4 ⊢ (∃𝑦 ∈ ran 𝐹𝜓 ↔ ∃𝑤 ∈ ran 𝐹[𝑤 / 𝑦]𝜓)
10	9	bicomi 132	. . 3 ⊢ (∃𝑤 ∈ ran 𝐹[𝑤 / 𝑦]𝜓 ↔ ∃𝑦 ∈ ran 𝐹𝜓)
11		nfmpt1 4122	. . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
12	1, 11	nfcxfr 2333	. . . . . 6 ⊢ Ⅎ𝑥𝐹
13		nfcv 2336	. . . . . 6 ⊢ Ⅎ𝑥𝑧
14	12, 13	nffv 5564	. . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑧)
15		nfv 1539	. . . . 5 ⊢ Ⅎ𝑥𝜓
16	14, 15	nfsbc 3006	. . . 4 ⊢ Ⅎ𝑥[(𝐹‘𝑧) / 𝑦]𝜓
17		nfv 1539	. . . 4 ⊢ Ⅎ𝑧[(𝐹‘𝑥) / 𝑦]𝜓
18		fveq2 5554	. . . . 5 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥))
19	18	sbceq1d 2990	. . . 4 ⊢ (𝑧 = 𝑥 → ([(𝐹‘𝑧) / 𝑦]𝜓 ↔ [(𝐹‘𝑥) / 𝑦]𝜓))
20	16, 17, 19	cbvrex 2723	. . 3 ⊢ (∃𝑧 ∈ 𝐴 [(𝐹‘𝑧) / 𝑦]𝜓 ↔ ∃𝑥 ∈ 𝐴 [(𝐹‘𝑥) / 𝑦]𝜓)
21	5, 10, 20	3bitr3g 222	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∃𝑦 ∈ ran 𝐹𝜓 ↔ ∃𝑥 ∈ 𝐴 [(𝐹‘𝑥) / 𝑦]𝜓))
22	1	fvmpt2 5641	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → (𝐹‘𝑥) = 𝐵)
23	22	sbceq1d 2990	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ [𝐵 / 𝑦]𝜓))
24		ralrnmpt.2	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
25	24	sbcieg 3018	. . . . . 6 ⊢ (𝐵 ∈ 𝑉 → ([𝐵 / 𝑦]𝜓 ↔ 𝜒))
26	25	adantl 277	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ([𝐵 / 𝑦]𝜓 ↔ 𝜒))
27	23, 26	bitrd 188	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ 𝜒))
28	27	ralimiaa 2556	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ 𝜒))
29		pm5.32 453	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 → ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ 𝜒)) ↔ ((𝑥 ∈ 𝐴 ∧ [(𝐹‘𝑥) / 𝑦]𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒)))
30	29	albii 1481	. . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ 𝜒)) ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ [(𝐹‘𝑥) / 𝑦]𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒)))
31		exbi 1615	. . . . 5 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ [(𝐹‘𝑥) / 𝑦]𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒)) → (∃𝑥(𝑥 ∈ 𝐴 ∧ [(𝐹‘𝑥) / 𝑦]𝜓) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜒)))
32	30, 31	sylbi 121	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ 𝜒)) → (∃𝑥(𝑥 ∈ 𝐴 ∧ [(𝐹‘𝑥) / 𝑦]𝜓) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜒)))
33		df-ral 2477	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ 𝜒) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ 𝜒)))
34		df-rex 2478	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 [(𝐹‘𝑥) / 𝑦]𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ [(𝐹‘𝑥) / 𝑦]𝜓))
35		df-rex 2478	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝜒 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜒))
36	34, 35	bibi12i 229	. . . 4 ⊢ ((∃𝑥 ∈ 𝐴 [(𝐹‘𝑥) / 𝑦]𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ [(𝐹‘𝑥) / 𝑦]𝜓) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜒)))
37	32, 33, 36	3imtr4i 201	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ 𝜒) → (∃𝑥 ∈ 𝐴 [(𝐹‘𝑥) / 𝑦]𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒))
38	28, 37	syl 14	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∃𝑥 ∈ 𝐴 [(𝐹‘𝑥) / 𝑦]𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒))
39	21, 38	bitrd 188	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∃𝑦 ∈ ran 𝐹𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1362   = wceq 1364  ∃wex 1503   ∈ wcel 2164  ∀wral 2472  ∃wrex 2473  [wsbc 2985   ↦ cmpt 4090  ran crn 4660   Fn wfn 5249  ‘cfv 5254

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-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262

This theorem is referenced by:  txbas  14426