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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 7162 | . 2 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
2 | elovimad.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐶) | |
3 | elovimad.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐷) | |
4 | 2, 3 | opelxpd 5596 | . . 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 3971 | . . . 4 ⊢ (𝜑 → 〈𝐴, 𝐵〉 ∈ dom 𝐹) |
8 | funfvima 6995 | . . . 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 2920 | 1 ⊢ (𝜑 → (𝐴𝐹𝐵) ∈ (𝐹 “ (𝐶 × 𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2113 ⊆ wss 3939 〈cop 4576 × cxp 5556 dom cdm 5558 “ cima 5561 Fun wfun 6352 ‘cfv 6358 (class class class)co 7159 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-fv 6366 df-ov 7162 |
This theorem is referenced by: ltgov 26386 xrofsup 30495 icoreelrn 34646 |
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