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Mirrors > Home > ILE Home > Th. List > ralima | GIF version |
Description: Universal quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.) |
Ref | Expression |
---|---|
rexima.x | ⊢ (𝑥 = (𝐹‘𝑦) → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
ralima | ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∀𝑥 ∈ (𝐹 “ 𝐵)𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel2 3165 | . . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐴) | |
2 | funfvex 5547 | . . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) → (𝐹‘𝑦) ∈ V) | |
3 | 2 | funfni 5331 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ V) |
4 | 1, 3 | sylan2 286 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 ⊆ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘𝑦) ∈ V) |
5 | 4 | anassrs 400 | . 2 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝐹‘𝑦) ∈ V) |
6 | fvelimab 5588 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑥 ∈ (𝐹 “ 𝐵) ↔ ∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑥)) | |
7 | eqcom 2191 | . . . 4 ⊢ ((𝐹‘𝑦) = 𝑥 ↔ 𝑥 = (𝐹‘𝑦)) | |
8 | 7 | rexbii 2497 | . . 3 ⊢ (∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑥 ↔ ∃𝑦 ∈ 𝐵 𝑥 = (𝐹‘𝑦)) |
9 | 6, 8 | bitrdi 196 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑥 ∈ (𝐹 “ 𝐵) ↔ ∃𝑦 ∈ 𝐵 𝑥 = (𝐹‘𝑦))) |
10 | rexima.x | . . 3 ⊢ (𝑥 = (𝐹‘𝑦) → (𝜑 ↔ 𝜓)) | |
11 | 10 | adantl 277 | . 2 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 = (𝐹‘𝑦)) → (𝜑 ↔ 𝜓)) |
12 | 5, 9, 11 | ralxfr2d 4479 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∀𝑥 ∈ (𝐹 “ 𝐵)𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1364 ∈ wcel 2160 ∀wral 2468 ∃wrex 2469 Vcvv 2752 ⊆ wss 3144 “ cima 4644 Fn wfn 5226 ‘cfv 5231 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-sbc 2978 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-fv 5239 |
This theorem is referenced by: supisolem 7025 mhmima 12909 ghmnsgima 13168 qtopbasss 14405 |
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