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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 7361 | . 2 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩) | |
2 | elovimad.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐶) | |
3 | elovimad.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐷) | |
4 | 2, 3 | opelxpd 5672 | . . 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 3946 | . . . 4 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ dom 𝐹) |
8 | funfvima 7181 | . . . 4 ⊢ ((Fun 𝐹 ∧ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹) → (⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) → (𝐹‘⟨𝐴, 𝐵⟩) ∈ (𝐹 “ (𝐶 × 𝐷)))) | |
9 | 5, 7, 8 | syl2anc 585 | . . 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 2842 | 1 ⊢ (𝜑 → (𝐴𝐹𝐵) ∈ (𝐹 “ (𝐶 × 𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 ⊆ wss 3911 ⟨cop 4593 × cxp 5632 dom cdm 5634 “ cima 5637 Fun wfun 6491 ‘cfv 6497 (class class class)co 7358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-fv 6505 df-ov 7361 |
This theorem is referenced by: ltgov 27542 xrofsup 31675 icoreelrn 35835 |
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