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Mirrors > Home > MPE Home > Th. List > ralima | Structured version Visualization version 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 | fvexd 6789 | . 2 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝐹‘𝑦) ∈ V) | |
2 | fvelimab 6841 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑥 ∈ (𝐹 “ 𝐵) ↔ ∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑥)) | |
3 | eqcom 2745 | . . . 4 ⊢ ((𝐹‘𝑦) = 𝑥 ↔ 𝑥 = (𝐹‘𝑦)) | |
4 | 3 | rexbii 3181 | . . 3 ⊢ (∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑥 ↔ ∃𝑦 ∈ 𝐵 𝑥 = (𝐹‘𝑦)) |
5 | 2, 4 | bitrdi 287 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑥 ∈ (𝐹 “ 𝐵) ↔ ∃𝑦 ∈ 𝐵 𝑥 = (𝐹‘𝑦))) |
6 | rexima.x | . . 3 ⊢ (𝑥 = (𝐹‘𝑦) → (𝜑 ↔ 𝜓)) | |
7 | 6 | adantl 482 | . 2 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 = (𝐹‘𝑦)) → (𝜑 ↔ 𝜓)) |
8 | 1, 5, 7 | ralxfr2d 5333 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∀𝑥 ∈ (𝐹 “ 𝐵)𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ∃wrex 3065 Vcvv 3432 ⊆ wss 3887 “ cima 5592 Fn wfn 6428 ‘cfv 6433 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-fv 6441 |
This theorem is referenced by: supisolem 9232 ordtypelem6 9282 ordtypelem7 9283 limsupgle 15186 mrcuni 17330 ipodrsima 18259 mhmima 18463 ghmnsgima 18858 cntzmhm 18945 qtopeu 22867 kqdisj 22883 ghmcnp 23266 qustgplem 23272 qtopbaslem 23922 bndth 24121 fmcfil 24436 ovoliunlem1 24666 volsup2 24769 mbflimsup 24830 itg2gt0 24925 mdegleb 25229 efopn 25813 fsumdvdsmul 26344 imaelshi 30420 cvmopnlem 33240 ovoliunnfl 35819 voliunnfl 35821 volsupnfl 35822 gicabl 40924 mgmhmima 45356 |
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