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Mirrors > Home > MPE Home > Th. List > ralrnmptw | Structured version Visualization version GIF version |
Description: A restricted quantifier over an image set. Version of ralrnmpt 7116 with a disjoint variable condition, which does not require ax-13 2375. (Contributed by Mario Carneiro, 20-Aug-2015.) Avoid ax-13 2375. (Revised by GG, 26-Jan-2024.) |
Ref | Expression |
---|---|
ralrnmptw.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
ralrnmptw.2 | ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
ralrnmptw | ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralrnmptw.1 | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
2 | 1 | fnmpt 6709 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → 𝐹 Fn 𝐴) |
3 | dfsbcq 3793 | . . . . 5 ⊢ (𝑤 = (𝐹‘𝑧) → ([𝑤 / 𝑦]𝜓 ↔ [(𝐹‘𝑧) / 𝑦]𝜓)) | |
4 | 3 | ralrn 7108 | . . . 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 | nfsbc1v 3811 | . . . 4 ⊢ Ⅎ𝑦[𝑤 / 𝑦]𝜓 | |
7 | nfv 1912 | . . . 4 ⊢ Ⅎ𝑤𝜓 | |
8 | sbceq2a 3803 | . . . 4 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑦]𝜓 ↔ 𝜓)) | |
9 | 6, 7, 8 | cbvralw 3304 | . . 3 ⊢ (∀𝑤 ∈ ran 𝐹[𝑤 / 𝑦]𝜓 ↔ ∀𝑦 ∈ ran 𝐹𝜓) |
10 | nfmpt1 5256 | . . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵) | |
11 | 1, 10 | nfcxfr 2901 | . . . . . 6 ⊢ Ⅎ𝑥𝐹 |
12 | nfcv 2903 | . . . . . 6 ⊢ Ⅎ𝑥𝑧 | |
13 | 11, 12 | nffv 6917 | . . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑧) |
14 | nfv 1912 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
15 | 13, 14 | nfsbcw 3813 | . . . 4 ⊢ Ⅎ𝑥[(𝐹‘𝑧) / 𝑦]𝜓 |
16 | nfv 1912 | . . . 4 ⊢ Ⅎ𝑧[(𝐹‘𝑥) / 𝑦]𝜓 | |
17 | fveq2 6907 | . . . . 5 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥)) | |
18 | 17 | sbceq1d 3796 | . . . 4 ⊢ (𝑧 = 𝑥 → ([(𝐹‘𝑧) / 𝑦]𝜓 ↔ [(𝐹‘𝑥) / 𝑦]𝜓)) |
19 | 15, 16, 18 | cbvralw 3304 | . . 3 ⊢ (∀𝑧 ∈ 𝐴 [(𝐹‘𝑧) / 𝑦]𝜓 ↔ ∀𝑥 ∈ 𝐴 [(𝐹‘𝑥) / 𝑦]𝜓) |
20 | 5, 9, 19 | 3bitr3g 313 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥 ∈ 𝐴 [(𝐹‘𝑥) / 𝑦]𝜓)) |
21 | 1 | fvmpt2 7027 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → (𝐹‘𝑥) = 𝐵) |
22 | 21 | sbceq1d 3796 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ [𝐵 / 𝑦]𝜓)) |
23 | ralrnmptw.2 | . . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
24 | 23 | sbcieg 3832 | . . . . . 6 ⊢ (𝐵 ∈ 𝑉 → ([𝐵 / 𝑦]𝜓 ↔ 𝜒)) |
25 | 24 | adantl 481 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ([𝐵 / 𝑦]𝜓 ↔ 𝜒)) |
26 | 22, 25 | bitrd 279 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ 𝜒)) |
27 | 26 | ralimiaa 3080 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ 𝜒)) |
28 | ralbi 3101 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ([(𝐹‘𝑥) / 𝑦]𝜓 ↔ 𝜒) → (∀𝑥 ∈ 𝐴 [(𝐹‘𝑥) / 𝑦]𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒)) | |
29 | 27, 28 | syl 17 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∀𝑥 ∈ 𝐴 [(𝐹‘𝑥) / 𝑦]𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒)) |
30 | 20, 29 | bitrd 279 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 [wsbc 3791 ↦ cmpt 5231 ran crn 5690 Fn wfn 6558 ‘cfv 6563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-fv 6571 |
This theorem is referenced by: rexrnmptw 7115 ac6num 10517 gsumwspan 18872 dfod2 19597 ordtbaslem 23212 ordtrest2lem 23227 cncmp 23416 comppfsc 23556 ptpjopn 23636 ordthmeolem 23825 tsmsfbas 24152 tsmsf1o 24169 prdsxmetlem 24394 prdsbl 24520 metdsf 24884 metdsge 24885 minveclem1 25472 minveclem3b 25476 minveclem6 25482 mbflimsup 25715 xrlimcnp 27026 minvecolem1 30903 minvecolem5 30910 minvecolem6 30911 ordtrest2NEWlem 33883 cvmsss2 35259 fin2so 37594 prdsbnd 37780 rrnequiv 37822 ralrnmpt3 45204 |
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