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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dmmptdf | Structured version Visualization version GIF version |
Description: The domain of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
dmmptdf.x | ⊢ Ⅎ𝑥𝜑 |
dmmptdf.a | ⊢ 𝐴 = (𝑥 ∈ 𝐵 ↦ 𝐶) |
dmmptdf.c | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝑉) |
Ref | Expression |
---|---|
dmmptdf | ⊢ (𝜑 → dom 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmmptdf.a | . . 3 ⊢ 𝐴 = (𝑥 ∈ 𝐵 ↦ 𝐶) | |
2 | 1 | dmmpt 6061 | . 2 ⊢ dom 𝐴 = {𝑥 ∈ 𝐵 ∣ 𝐶 ∈ V} |
3 | dmmptdf.x | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
4 | dmmptdf.c | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝑉) | |
5 | 4 | elexd 3461 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ V) |
6 | 3, 5 | ralrimia 41767 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝐶 ∈ V) |
7 | rabid2 3334 | . . 3 ⊢ (𝐵 = {𝑥 ∈ 𝐵 ∣ 𝐶 ∈ V} ↔ ∀𝑥 ∈ 𝐵 𝐶 ∈ V) | |
8 | 6, 7 | sylibr 237 | . 2 ⊢ (𝜑 → 𝐵 = {𝑥 ∈ 𝐵 ∣ 𝐶 ∈ V}) |
9 | 2, 8 | eqtr4id 2852 | 1 ⊢ (𝜑 → dom 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 Ⅎwnf 1785 ∈ wcel 2111 ∀wral 3106 {crab 3110 Vcvv 3441 ↦ cmpt 5110 dom cdm 5519 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-mpt 5111 df-xp 5525 df-rel 5526 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 |
This theorem is referenced by: smfpimltmpt 43380 smfpimltxrmpt 43392 smfadd 43398 smfpimgtmpt 43414 smfpimgtxrmpt 43417 smfpimioompt 43418 smfrec 43421 smfmul 43427 smfmulc1 43428 smffmpt 43436 smfsupmpt 43446 smfinfmpt 43450 smflimsupmpt 43460 smfliminfmpt 43463 |
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