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Mirrors > Home > MPE Home > Th. List > elovimad | Structured version Visualization version GIF version |
Description: Elementhood of the image set of an operation value. (Contributed by Thierry Arnoux, 13-Mar-2017.) |
Ref | Expression |
---|---|
elovimad.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐶) |
elovimad.2 | ⊢ (𝜑 → 𝐵 ∈ 𝐷) |
elovimad.3 | ⊢ (𝜑 → Fun 𝐹) |
elovimad.4 | ⊢ (𝜑 → (𝐶 × 𝐷) ⊆ dom 𝐹) |
Ref | Expression |
---|---|
elovimad | ⊢ (𝜑 → (𝐴𝐹𝐵) ∈ (𝐹 “ (𝐶 × 𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ov 7138 | . 2 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
2 | elovimad.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐶) | |
3 | elovimad.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐷) | |
4 | 2, 3 | opelxpd 5557 | . . 3 ⊢ (𝜑 → 〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷)) |
5 | elovimad.3 | . . . 4 ⊢ (𝜑 → Fun 𝐹) | |
6 | elovimad.4 | . . . . 5 ⊢ (𝜑 → (𝐶 × 𝐷) ⊆ dom 𝐹) | |
7 | 6, 4 | sseldd 3916 | . . . 4 ⊢ (𝜑 → 〈𝐴, 𝐵〉 ∈ dom 𝐹) |
8 | funfvima 6970 | . . . 4 ⊢ ((Fun 𝐹 ∧ 〈𝐴, 𝐵〉 ∈ dom 𝐹) → (〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷) → (𝐹‘〈𝐴, 𝐵〉) ∈ (𝐹 “ (𝐶 × 𝐷)))) | |
9 | 5, 7, 8 | syl2anc 587 | . . 3 ⊢ (𝜑 → (〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷) → (𝐹‘〈𝐴, 𝐵〉) ∈ (𝐹 “ (𝐶 × 𝐷)))) |
10 | 4, 9 | mpd 15 | . 2 ⊢ (𝜑 → (𝐹‘〈𝐴, 𝐵〉) ∈ (𝐹 “ (𝐶 × 𝐷))) |
11 | 1, 10 | eqeltrid 2894 | 1 ⊢ (𝜑 → (𝐴𝐹𝐵) ∈ (𝐹 “ (𝐶 × 𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 ⊆ wss 3881 〈cop 4531 × cxp 5517 dom cdm 5519 “ cima 5522 Fun wfun 6318 ‘cfv 6324 (class class class)co 7135 |
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-rex 3112 df-v 3443 df-sbc 3721 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-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-fv 6332 df-ov 7138 |
This theorem is referenced by: ltgov 26391 xrofsup 30518 icoreelrn 34778 |
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