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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > imassmpt | Structured version Visualization version GIF version |
Description: Membership relation for the values of a function whose image is a subclass. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
imassmpt.1 | ⊢ Ⅎ𝑥𝜑 |
imassmpt.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐶)) → 𝐵 ∈ 𝑉) |
imassmpt.3 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
Ref | Expression |
---|---|
imassmpt | ⊢ (𝜑 → ((𝐹 “ 𝐶) ⊆ 𝐷 ↔ ∀𝑥 ∈ (𝐴 ∩ 𝐶)𝐵 ∈ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ima 5370 | . . . . 5 ⊢ (𝐹 “ 𝐶) = ran (𝐹 ↾ 𝐶) | |
2 | imassmpt.3 | . . . . . . . 8 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
3 | 2 | reseq1i 5640 | . . . . . . 7 ⊢ (𝐹 ↾ 𝐶) = ((𝑥 ∈ 𝐴 ↦ 𝐵) ↾ 𝐶) |
4 | resmpt3 5702 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) ↾ 𝐶) = (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵) | |
5 | 3, 4 | eqtri 2802 | . . . . . 6 ⊢ (𝐹 ↾ 𝐶) = (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵) |
6 | 5 | rneqi 5599 | . . . . 5 ⊢ ran (𝐹 ↾ 𝐶) = ran (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵) |
7 | 1, 6 | eqtri 2802 | . . . 4 ⊢ (𝐹 “ 𝐶) = ran (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵) |
8 | 7 | sseq1i 3848 | . . 3 ⊢ ((𝐹 “ 𝐶) ⊆ 𝐷 ↔ ran (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵) ⊆ 𝐷) |
9 | 8 | a1i 11 | . 2 ⊢ (𝜑 → ((𝐹 “ 𝐶) ⊆ 𝐷 ↔ ran (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵) ⊆ 𝐷)) |
10 | imassmpt.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
11 | eqid 2778 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵) = (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵) | |
12 | imassmpt.2 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐶)) → 𝐵 ∈ 𝑉) | |
13 | 10, 11, 12 | rnmptssbi 40398 | . 2 ⊢ (𝜑 → (ran (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵) ⊆ 𝐷 ↔ ∀𝑥 ∈ (𝐴 ∩ 𝐶)𝐵 ∈ 𝐷)) |
14 | 9, 13 | bitrd 271 | 1 ⊢ (𝜑 → ((𝐹 “ 𝐶) ⊆ 𝐷 ↔ ∀𝑥 ∈ (𝐴 ∩ 𝐶)𝐵 ∈ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 Ⅎwnf 1827 ∈ wcel 2107 ∀wral 3090 ∩ cin 3791 ⊆ wss 3792 ↦ cmpt 4967 ran crn 5358 ↾ cres 5359 “ cima 5360 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pr 5140 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-fv 6145 |
This theorem is referenced by: limsup10exlem 40922 |
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