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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 6653 | . 2 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
2 | elovimad.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐶) | |
3 | elovimad.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐷) | |
4 | opelxpi 5148 | . . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷)) | |
5 | 2, 3, 4 | syl2anc 693 | . . 3 ⊢ (𝜑 → 〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷)) |
6 | elovimad.3 | . . . 4 ⊢ (𝜑 → Fun 𝐹) | |
7 | elovimad.4 | . . . . 5 ⊢ (𝜑 → (𝐶 × 𝐷) ⊆ dom 𝐹) | |
8 | 7, 5 | sseldd 3604 | . . . 4 ⊢ (𝜑 → 〈𝐴, 𝐵〉 ∈ dom 𝐹) |
9 | funfvima 6492 | . . . 4 ⊢ ((Fun 𝐹 ∧ 〈𝐴, 𝐵〉 ∈ dom 𝐹) → (〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷) → (𝐹‘〈𝐴, 𝐵〉) ∈ (𝐹 “ (𝐶 × 𝐷)))) | |
10 | 6, 8, 9 | syl2anc 693 | . . 3 ⊢ (𝜑 → (〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷) → (𝐹‘〈𝐴, 𝐵〉) ∈ (𝐹 “ (𝐶 × 𝐷)))) |
11 | 5, 10 | mpd 15 | . 2 ⊢ (𝜑 → (𝐹‘〈𝐴, 𝐵〉) ∈ (𝐹 “ (𝐶 × 𝐷))) |
12 | 1, 11 | syl5eqel 2705 | 1 ⊢ (𝜑 → (𝐴𝐹𝐵) ∈ (𝐹 “ (𝐶 × 𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 ⊆ wss 3574 〈cop 4183 × cxp 5112 dom cdm 5114 “ cima 5117 Fun wfun 5882 ‘cfv 5888 (class class class)co 6650 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-fv 5896 df-ov 6653 |
This theorem is referenced by: ltgov 25492 xrofsup 29533 icoreelrn 33209 |
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