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Mirrors > Home > MPE Home > Th. List > Mathboxes > elrnmpoid | Structured version Visualization version GIF version |
Description: Membership in the range of an operation class abstraction. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
elrnmpoid.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) |
Ref | Expression |
---|---|
elrnmpoid | ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉) → (𝑥𝐹𝑦) ∈ ran 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elrnmpoid.1 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) | |
2 | 1 | fnmpo 8057 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → 𝐹 Fn (𝐴 × 𝐵)) |
3 | 2 | 3ad2ant3 1135 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉) → 𝐹 Fn (𝐴 × 𝐵)) |
4 | simp1 1136 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉) → 𝑥 ∈ 𝐴) | |
5 | simp2 1137 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉) → 𝑦 ∈ 𝐵) | |
6 | fnovrn 7584 | . 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥𝐹𝑦) ∈ ran 𝐹) | |
7 | 3, 4, 5, 6 | syl3anc 1371 | 1 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉) → (𝑥𝐹𝑦) ∈ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3061 × cxp 5674 ran crn 5677 Fn wfn 6538 (class class class)co 7411 ∈ cmpo 7413 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-ov 7414 df-oprab 7415 df-mpo 7416 df-1st 7977 df-2nd 7978 |
This theorem is referenced by: smflimlem1 45566 smflimlem2 45567 |
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