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| Mirrors > Home > MPE Home > Th. List > Mathboxes > wfaxrep | Structured version Visualization version GIF version | ||
| Description: The class of well-founded sets models the Axiom of Replacement ax-rep 5277. Actually, our statement is stronger, since it is an instance of Replacement only when all quantifiers in ∀𝑦𝜑 are relativized to 𝑊. Essentially part of Corollary II.2.5 of [Kunen2] p. 112, but note that our Replacement is different from Kunen's. (Contributed by Eric Schmidt, 29-Sep-2025.) |
| Ref | Expression |
|---|---|
| wfax.1 | ⊢ 𝑊 = ∪ (𝑅1 “ On) |
| Ref | Expression |
|---|---|
| wfaxrep | ⊢ ∀𝑥 ∈ 𝑊 (∀𝑤 ∈ 𝑊 ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 (∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑊 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | wfax.1 | . 2 ⊢ 𝑊 = ∪ (𝑅1 “ On) | |
| 2 | trwf 44954 | . . . 4 ⊢ Tr ∪ (𝑅1 “ On) | |
| 3 | treq 5265 | . . . 4 ⊢ (𝑊 = ∪ (𝑅1 “ On) → (Tr 𝑊 ↔ Tr ∪ (𝑅1 “ On))) | |
| 4 | 2, 3 | mpbiri 258 | . . 3 ⊢ (𝑊 = ∪ (𝑅1 “ On) → Tr 𝑊) |
| 5 | vex 3483 | . . . . . . . . . 10 ⊢ 𝑓 ∈ V | |
| 6 | 5 | rnex 7928 | . . . . . . . . 9 ⊢ ran 𝑓 ∈ V |
| 7 | 6 | r1elss 9842 | . . . . . . . 8 ⊢ (ran 𝑓 ∈ ∪ (𝑅1 “ On) ↔ ran 𝑓 ⊆ ∪ (𝑅1 “ On)) |
| 8 | 7 | biimpri 228 | . . . . . . 7 ⊢ (ran 𝑓 ⊆ ∪ (𝑅1 “ On) → ran 𝑓 ∈ ∪ (𝑅1 “ On)) |
| 9 | 1 | sseq2i 4012 | . . . . . . 7 ⊢ (ran 𝑓 ⊆ 𝑊 ↔ ran 𝑓 ⊆ ∪ (𝑅1 “ On)) |
| 10 | 1 | eleq2i 2832 | . . . . . . 7 ⊢ (ran 𝑓 ∈ 𝑊 ↔ ran 𝑓 ∈ ∪ (𝑅1 “ On)) |
| 11 | 8, 9, 10 | 3imtr4i 292 | . . . . . 6 ⊢ (ran 𝑓 ⊆ 𝑊 → ran 𝑓 ∈ 𝑊) |
| 12 | 11 | 3ad2ant3 1136 | . . . . 5 ⊢ ((Fun 𝑓 ∧ dom 𝑓 ∈ 𝑊 ∧ ran 𝑓 ⊆ 𝑊) → ran 𝑓 ∈ 𝑊) |
| 13 | 12 | ax-gen 1795 | . . . 4 ⊢ ∀𝑓((Fun 𝑓 ∧ dom 𝑓 ∈ 𝑊 ∧ ran 𝑓 ⊆ 𝑊) → ran 𝑓 ∈ 𝑊) |
| 14 | 13 | a1i 11 | . . 3 ⊢ (𝑊 = ∪ (𝑅1 “ On) → ∀𝑓((Fun 𝑓 ∧ dom 𝑓 ∈ 𝑊 ∧ ran 𝑓 ⊆ 𝑊) → ran 𝑓 ∈ 𝑊)) |
| 15 | onwf 9866 | . . . . 5 ⊢ On ⊆ ∪ (𝑅1 “ On) | |
| 16 | 0elon 6436 | . . . . 5 ⊢ ∅ ∈ On | |
| 17 | 15, 16 | sselii 3979 | . . . 4 ⊢ ∅ ∈ ∪ (𝑅1 “ On) |
| 18 | eleq2 2829 | . . . 4 ⊢ (𝑊 = ∪ (𝑅1 “ On) → (∅ ∈ 𝑊 ↔ ∅ ∈ ∪ (𝑅1 “ On))) | |
| 19 | 17, 18 | mpbiri 258 | . . 3 ⊢ (𝑊 = ∪ (𝑅1 “ On) → ∅ ∈ 𝑊) |
| 20 | 4, 14, 19 | modelaxrep 44974 | . 2 ⊢ (𝑊 = ∪ (𝑅1 “ On) → ∀𝑥 ∈ 𝑊 (∀𝑤 ∈ 𝑊 ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 (∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑊 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))) |
| 21 | 1, 20 | ax-mp 5 | 1 ⊢ ∀𝑥 ∈ 𝑊 (∀𝑤 ∈ 𝑊 ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 (∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑊 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 ∀wal 1538 = wceq 1540 ∈ wcel 2108 ∀wral 3060 ∃wrex 3069 ⊆ wss 3950 ∅c0 4332 ∪ cuni 4905 Tr wtr 5257 dom cdm 5683 ran crn 5684 “ cima 5686 Oncon0 6382 Fun wfun 6553 𝑅1cr1 9798 |
| 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-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-int 4945 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5224 df-tr 5258 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-we 5637 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-pred 6319 df-ord 6385 df-on 6386 df-lim 6387 df-suc 6388 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-ov 7432 df-om 7884 df-2nd 8011 df-frecs 8302 df-wrecs 8333 df-recs 8407 df-rdg 8446 df-en 8982 df-dom 8983 df-sdom 8984 df-r1 9800 df-rank 9801 |
| This theorem is referenced by: (None) |
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