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Mirrors > Home > ILE Home > Th. List > funimassov | 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 5167 | . 2 ⊢ ((Fun 𝐹 ∧ (A × B) ⊆ dom 𝐹) → ((𝐹 “ (A × B)) ⊆ 𝐶 ↔ ∀z ∈ (A × B)(𝐹‘z) ∈ 𝐶)) | |
2 | fveq2 5121 | . . . . 5 ⊢ (z = 〈x, y〉 → (𝐹‘z) = (𝐹‘〈x, y〉)) | |
3 | df-ov 5458 | . . . . 5 ⊢ (x𝐹y) = (𝐹‘〈x, y〉) | |
4 | 2, 3 | syl6eqr 2087 | . . . 4 ⊢ (z = 〈x, y〉 → (𝐹‘z) = (x𝐹y)) |
5 | 4 | eleq1d 2103 | . . 3 ⊢ (z = 〈x, y〉 → ((𝐹‘z) ∈ 𝐶 ↔ (x𝐹y) ∈ 𝐶)) |
6 | 5 | ralxp 4422 | . 2 ⊢ (∀z ∈ (A × B)(𝐹‘z) ∈ 𝐶 ↔ ∀x ∈ A ∀y ∈ B (x𝐹y) ∈ 𝐶) |
7 | 1, 6 | syl6bb 185 | 1 ⊢ ((Fun 𝐹 ∧ (A × B) ⊆ dom 𝐹) → ((𝐹 “ (A × B)) ⊆ 𝐶 ↔ ∀x ∈ A ∀y ∈ B (x𝐹y) ∈ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1242 ∈ wcel 1390 ∀wral 2300 ⊆ wss 2911 〈cop 3370 × cxp 4286 dom cdm 4288 “ cima 4291 Fun wfun 4839 ‘cfv 4845 (class class class)co 5455 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-sbc 2759 df-csb 2847 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-iun 3650 df-br 3756 df-opab 3810 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-fv 4853 df-ov 5458 |
This theorem is referenced by: (None) |
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