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Mirrors > Home > MPE Home > Th. List > ralrnmpt | Structured version Visualization version GIF version |
Description: A restricted quantifier over an image set. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
ralrnmpt.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
ralrnmpt.2 | ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
ralrnmpt | ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralrnmpt.1 | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
2 | 1 | fnmpt 5933 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → 𝐹 Fn 𝐴) |
3 | dfsbcq 3404 | . . . . 5 ⊢ (𝑤 = (𝐹‘𝑧) → ([𝑤 / 𝑦]𝜓 ↔ [(𝐹‘𝑧) / 𝑦]𝜓)) | |
4 | 3 | ralrn 6270 | . . . 4 ⊢ (𝐹 Fn 𝐴 → (∀𝑤 ∈ ran 𝐹[𝑤 / 𝑦]𝜓 ↔ ∀𝑧 ∈ 𝐴 [(𝐹‘𝑧) / 𝑦]𝜓)) |
5 | 2, 4 | syl 17 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∀𝑤 ∈ ran 𝐹[𝑤 / 𝑦]𝜓 ↔ ∀𝑧 ∈ 𝐴 [(𝐹‘𝑧) / 𝑦]𝜓)) |
6 | nfv 1830 | . . . . 5 ⊢ Ⅎ𝑤𝜓 | |
7 | nfsbc1v 3422 | . . . . 5 ⊢ Ⅎ𝑦[𝑤 / 𝑦]𝜓 | |
8 | sbceq1a 3413 | . . . . 5 ⊢ (𝑦 = 𝑤 → (𝜓 ↔ [𝑤 / 𝑦]𝜓)) | |
9 | 6, 7, 8 | cbvral 3143 | . . . 4 ⊢ (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑤 ∈ ran 𝐹[𝑤 / 𝑦]𝜓) |
10 | 9 | bicomi 213 | . . 3 ⊢ (∀𝑤 ∈ ran 𝐹[𝑤 / 𝑦]𝜓 ↔ ∀𝑦 ∈ ran 𝐹𝜓) |
11 | nfmpt1 4675 | . . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵) | |
12 | 1, 11 | nfcxfr 2749 | . . . . . 6 ⊢ Ⅎ𝑥𝐹 |
13 | nfcv 2751 | . . . . . 6 ⊢ Ⅎ𝑥𝑧 | |
14 | 12, 13 | nffv 6110 | . . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑧) |
15 | nfv 1830 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
16 | 14, 15 | nfsbc 3424 | . . . 4 ⊢ Ⅎ𝑥[(𝐹‘𝑧) / 𝑦]𝜓 |
17 | nfv 1830 | . . . 4 ⊢ Ⅎ𝑧[(𝐹‘𝑥) / 𝑦]𝜓 | |
18 | fveq2 6103 | . . . . 5 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥)) | |
19 | 18 | sbceq1d 3407 | . . . 4 ⊢ (𝑧 = 𝑥 → ([(𝐹‘𝑧) / 𝑦]𝜓 ↔ [(𝐹‘𝑥) / 𝑦]𝜓)) |
20 | 16, 17, 19 | cbvral 3143 | . . 3 ⊢ (∀𝑧 ∈ 𝐴 [(𝐹‘𝑧) / 𝑦]𝜓 ↔ ∀𝑥 ∈ 𝐴 [(𝐹‘𝑥) / 𝑦]𝜓) |
21 | 5, 10, 20 | 3bitr3g 301 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥 ∈ 𝐴 [(𝐹‘𝑥) / 𝑦]𝜓)) |
22 | 1 | fvmpt2 6200 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → (𝐹‘𝑥) = 𝐵) |
23 | 22 | sbceq1d 3407 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ [𝐵 / 𝑦]𝜓)) |
24 | ralrnmpt.2 | . . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
25 | 24 | sbcieg 3435 | . . . . . 6 ⊢ (𝐵 ∈ 𝑉 → ([𝐵 / 𝑦]𝜓 ↔ 𝜒)) |
26 | 25 | adantl 481 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ([𝐵 / 𝑦]𝜓 ↔ 𝜒)) |
27 | 23, 26 | bitrd 267 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ 𝜒)) |
28 | 27 | ralimiaa 2935 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ 𝜒)) |
29 | ralbi 3050 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ 𝜒) → (∀𝑥 ∈ 𝐴 [(𝐹‘𝑥) / 𝑦]𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒)) | |
30 | 28, 29 | syl 17 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∀𝑥 ∈ 𝐴 [(𝐹‘𝑥) / 𝑦]𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒)) |
31 | 21, 30 | bitrd 267 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 [wsbc 3402 ↦ cmpt 4643 ran crn 5039 Fn wfn 5799 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-fv 5812 |
This theorem is referenced by: rexrnmpt 6277 ac6num 9184 gsumwspan 17206 dfod2 17804 ordtbaslem 20802 ordtrest2lem 20817 cncmp 21005 comppfsc 21145 ptpjopn 21225 ordthmeolem 21414 tsmsfbas 21741 tsmsf1o 21758 prdsxmetlem 21983 prdsbl 22106 metdsf 22459 metdsge 22460 minveclem1 23003 minveclem3b 23007 minveclem6 23013 mbflimsup 23239 xrlimcnp 24495 minvecolem1 27114 minvecolem5 27121 minvecolem6 27122 ordtrest2NEWlem 29296 cvmsss2 30510 fin2so 32566 prdsbnd 32762 rrnequiv 32804 |
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