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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 7153 | . 2 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
2 | elovimad.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐶) | |
3 | elovimad.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐷) | |
4 | 2, 3 | opelxpd 5587 | . . 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 3967 | . . . 4 ⊢ (𝜑 → 〈𝐴, 𝐵〉 ∈ dom 𝐹) |
8 | funfvima 6986 | . . . 4 ⊢ ((Fun 𝐹 ∧ 〈𝐴, 𝐵〉 ∈ dom 𝐹) → (〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷) → (𝐹‘〈𝐴, 𝐵〉) ∈ (𝐹 “ (𝐶 × 𝐷)))) | |
9 | 5, 7, 8 | syl2anc 586 | . . 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 2917 | 1 ⊢ (𝜑 → (𝐴𝐹𝐵) ∈ (𝐹 “ (𝐶 × 𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 ⊆ wss 3935 〈cop 4566 × cxp 5547 dom cdm 5549 “ cima 5552 Fun wfun 6343 ‘cfv 6349 (class class class)co 7150 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-fv 6357 df-ov 7153 |
This theorem is referenced by: ltgov 26377 xrofsup 30486 icoreelrn 34636 |
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