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Mirrors > Home > MPE Home > Th. List > ac6sg | Structured version Visualization version GIF version |
Description: ac6s 9894 with sethood as antecedent. (Contributed by FL, 3-Aug-2009.) |
Ref | Expression |
---|---|
ac6sg.1 | ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
ac6sg | ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | raleq 3403 | . . 3 ⊢ (𝑧 = 𝐴 → (∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)) | |
2 | feq2 6489 | . . . . 5 ⊢ (𝑧 = 𝐴 → (𝑓:𝑧⟶𝐵 ↔ 𝑓:𝐴⟶𝐵)) | |
3 | raleq 3403 | . . . . 5 ⊢ (𝑧 = 𝐴 → (∀𝑥 ∈ 𝑧 𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜓)) | |
4 | 2, 3 | anbi12d 630 | . . . 4 ⊢ (𝑧 = 𝐴 → ((𝑓:𝑧⟶𝐵 ∧ ∀𝑥 ∈ 𝑧 𝜓) ↔ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓))) |
5 | 4 | exbidv 1913 | . . 3 ⊢ (𝑧 = 𝐴 → (∃𝑓(𝑓:𝑧⟶𝐵 ∧ ∀𝑥 ∈ 𝑧 𝜓) ↔ ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓))) |
6 | 1, 5 | imbi12d 346 | . 2 ⊢ (𝑧 = 𝐴 → ((∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑓(𝑓:𝑧⟶𝐵 ∧ ∀𝑥 ∈ 𝑧 𝜓)) ↔ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓)))) |
7 | vex 3495 | . . 3 ⊢ 𝑧 ∈ V | |
8 | ac6sg.1 | . . 3 ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓)) | |
9 | 7, 8 | ac6s 9894 | . 2 ⊢ (∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑓(𝑓:𝑧⟶𝐵 ∧ ∀𝑥 ∈ 𝑧 𝜓)) |
10 | 6, 9 | vtoclg 3565 | 1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∃wex 1771 ∈ wcel 2105 ∀wral 3135 ∃wrex 3136 ⟶wf 6344 ‘cfv 6348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-reg 9044 ax-inf2 9092 ax-ac2 9873 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-en 8498 df-r1 9181 df-rank 9182 df-card 9356 df-ac 9530 |
This theorem is referenced by: acsmapd 17776 foresf1o 30192 reff 31002 cmpcref 31013 omssubadd 31457 nlpfvineqsn 34572 ac6gf 34888 |
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