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| Mirrors > Home > MPE Home > Th. List > funimassov | Structured version Visualization version GIF version | ||
| Description: Membership relation for the values of a function whose image is a subclass. (Contributed by Mario Carneiro, 23-Dec-2013.) |
| Ref | Expression |
|---|---|
| funimassov | ⊢ ((Fun 𝐹 ∧ (𝐴 × 𝐵) ⊆ dom 𝐹) → ((𝐹 “ (𝐴 × 𝐵)) ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) ∈ 𝐶)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | funimass4 6973 | . 2 ⊢ ((Fun 𝐹 ∧ (𝐴 × 𝐵) ⊆ dom 𝐹) → ((𝐹 “ (𝐴 × 𝐵)) ⊆ 𝐶 ↔ ∀𝑧 ∈ (𝐴 × 𝐵)(𝐹‘𝑧) ∈ 𝐶)) | |
| 2 | fveq2 6906 | . . . . 5 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝐹‘𝑧) = (𝐹‘〈𝑥, 𝑦〉)) | |
| 3 | df-ov 7434 | . . . . 5 ⊢ (𝑥𝐹𝑦) = (𝐹‘〈𝑥, 𝑦〉) | |
| 4 | 2, 3 | eqtr4di 2795 | . . . 4 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝐹‘𝑧) = (𝑥𝐹𝑦)) |
| 5 | 4 | eleq1d 2826 | . . 3 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → ((𝐹‘𝑧) ∈ 𝐶 ↔ (𝑥𝐹𝑦) ∈ 𝐶)) |
| 6 | 5 | ralxp 5852 | . 2 ⊢ (∀𝑧 ∈ (𝐴 × 𝐵)(𝐹‘𝑧) ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) ∈ 𝐶) |
| 7 | 1, 6 | bitrdi 287 | 1 ⊢ ((Fun 𝐹 ∧ (𝐴 × 𝐵) ⊆ dom 𝐹) → ((𝐹 “ (𝐴 × 𝐵)) ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) ∈ 𝐶)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ⊆ wss 3951 〈cop 4632 × cxp 5683 dom cdm 5685 “ cima 5688 Fun wfun 6555 ‘cfv 6561 (class class class)co 7431 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-fv 6569 df-ov 7434 |
| This theorem is referenced by: naddcllem 8714 naddov2 8717 naddasslem1 8732 naddasslem2 8733 dprd2da 20062 xkococnlem 23667 iscfil2 25300 itg1addlem4 25734 issh2 31228 cvmlift2lem9 35316 |
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