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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.) Reduce DV conditions. (Revised by Matthew House, 14-Aug-2025.) |
| Ref | Expression |
|---|---|
| ralima.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 | fnfun 6668 | . . 3 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹) | |
| 2 | 1 | funfnd 6597 | . 2 ⊢ (𝐹 Fn 𝐴 → 𝐹 Fn dom 𝐹) |
| 3 | fndm 6671 | . . . 4 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
| 4 | 3 | sseq2d 4016 | . . 3 ⊢ (𝐹 Fn 𝐴 → (𝐵 ⊆ dom 𝐹 ↔ 𝐵 ⊆ 𝐴)) |
| 5 | 4 | biimpar 477 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ dom 𝐹) |
| 6 | fvexd 6921 | . . 3 ⊢ (((𝐹 Fn dom 𝐹 ∧ 𝐵 ⊆ dom 𝐹) ∧ 𝑦 ∈ 𝐵) → (𝐹‘𝑦) ∈ V) | |
| 7 | fvelimab 6981 | . . . 4 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝐵 ⊆ dom 𝐹) → (𝑥 ∈ (𝐹 “ 𝐵) ↔ ∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑥)) | |
| 8 | eqcom 2744 | . . . . 5 ⊢ ((𝐹‘𝑦) = 𝑥 ↔ 𝑥 = (𝐹‘𝑦)) | |
| 9 | 8 | rexbii 3094 | . . . 4 ⊢ (∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑥 ↔ ∃𝑦 ∈ 𝐵 𝑥 = (𝐹‘𝑦)) |
| 10 | 7, 9 | bitrdi 287 | . . 3 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝐵 ⊆ dom 𝐹) → (𝑥 ∈ (𝐹 “ 𝐵) ↔ ∃𝑦 ∈ 𝐵 𝑥 = (𝐹‘𝑦))) |
| 11 | ralima.x | . . . 4 ⊢ (𝑥 = (𝐹‘𝑦) → (𝜑 ↔ 𝜓)) | |
| 12 | 11 | adantl 481 | . . 3 ⊢ (((𝐹 Fn dom 𝐹 ∧ 𝐵 ⊆ dom 𝐹) ∧ 𝑥 = (𝐹‘𝑦)) → (𝜑 ↔ 𝜓)) |
| 13 | 6, 10, 12 | ralxfr2d 5410 | . 2 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝐵 ⊆ dom 𝐹) → (∀𝑥 ∈ (𝐹 “ 𝐵)𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜓)) |
| 14 | 2, 5, 13 | syl2an2r 685 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∀𝑥 ∈ (𝐹 “ 𝐵)𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜓)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 Vcvv 3480 ⊆ wss 3951 dom cdm 5685 “ cima 5688 Fn wfn 6556 ‘cfv 6561 |
| 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-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-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 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-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 |
| This theorem is referenced by: rexima 7258 supisolem 9513 ordtypelem6 9563 ordtypelem7 9564 limsupgle 15513 mrcuni 17664 ipodrsima 18586 mgmhmima 18728 mhmimalem 18837 ghmnsgima 19258 cntzmhm 19359 rhmimasubrnglem 20565 qtopeu 23724 kqdisj 23740 ghmcnp 24123 qustgplem 24129 qtopbaslem 24779 bndth 24990 fmcfil 25306 ovoliunlem1 25537 volsup2 25640 mbflimsup 25701 itg2gt0 25795 mdegleb 26103 efopn 26700 fsumdvdsmul 27238 fsumdvdsmulOLD 27240 negsunif 28087 negsbdaylem 28088 imaelshi 32077 cvmopnlem 35283 weiunfrlem 36465 ovoliunnfl 37669 voliunnfl 37671 volsupnfl 37672 gicabl 43111 |
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