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Mirrors > Home > MPE Home > Th. List > Mathboxes > elmapssresd | Structured version Visualization version GIF version |
Description: A restricted mapping is a mapping. EDITORIAL: Could be used to shorten elpm2r 8834 with some reordering involving mapsspm 8865. (Contributed by SN, 11-Mar-2025.) |
Ref | Expression |
---|---|
elmapssresd.1 | ⊢ (𝜑 → 𝐴 ∈ (𝐵 ↑m 𝐶)) |
elmapssresd.2 | ⊢ (𝜑 → 𝐷 ⊆ 𝐶) |
Ref | Expression |
---|---|
elmapssresd | ⊢ (𝜑 → (𝐴 ↾ 𝐷) ∈ (𝐵 ↑m 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elmapssresd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ (𝐵 ↑m 𝐶)) | |
2 | elmapssresd.2 | . 2 ⊢ (𝜑 → 𝐷 ⊆ 𝐶) | |
3 | elmapssres 8856 | . 2 ⊢ ((𝐴 ∈ (𝐵 ↑m 𝐶) ∧ 𝐷 ⊆ 𝐶) → (𝐴 ↾ 𝐷) ∈ (𝐵 ↑m 𝐷)) | |
4 | 1, 2, 3 | syl2anc 583 | 1 ⊢ (𝜑 → (𝐴 ↾ 𝐷) ∈ (𝐵 ↑m 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 ⊆ wss 3940 ↾ cres 5668 (class class class)co 7401 ↑m cmap 8815 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-fv 6541 df-ov 7404 df-oprab 7405 df-mpo 7406 df-1st 7968 df-2nd 7969 df-map 8817 |
This theorem is referenced by: evlselv 41614 |
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