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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 5284. 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 44936 | . . . 4 ⊢ Tr ∪ (𝑅1 “ On) | |
3 | treq 5272 | . . . 4 ⊢ (𝑊 = ∪ (𝑅1 “ On) → (Tr 𝑊 ↔ Tr ∪ (𝑅1 “ On))) | |
4 | 2, 3 | mpbiri 258 | . . 3 ⊢ (𝑊 = ∪ (𝑅1 “ On) → Tr 𝑊) |
5 | vex 3481 | . . . . . . . . . 10 ⊢ 𝑓 ∈ V | |
6 | 5 | rnex 7932 | . . . . . . . . 9 ⊢ ran 𝑓 ∈ V |
7 | 6 | r1elss 9843 | . . . . . . . 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 4024 | . . . . . . 7 ⊢ (ran 𝑓 ⊆ 𝑊 ↔ ran 𝑓 ⊆ ∪ (𝑅1 “ On)) |
10 | 1 | eleq2i 2830 | . . . . . . 7 ⊢ (ran 𝑓 ∈ 𝑊 ↔ ran 𝑓 ∈ ∪ (𝑅1 “ On)) |
11 | 8, 9, 10 | 3imtr4i 292 | . . . . . 6 ⊢ (ran 𝑓 ⊆ 𝑊 → ran 𝑓 ∈ 𝑊) |
12 | 11 | 3ad2ant3 1134 | . . . . 5 ⊢ ((Fun 𝑓 ∧ dom 𝑓 ∈ 𝑊 ∧ ran 𝑓 ⊆ 𝑊) → ran 𝑓 ∈ 𝑊) |
13 | 12 | ax-gen 1791 | . . . 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 9867 | . . . . 5 ⊢ On ⊆ ∪ (𝑅1 “ On) | |
16 | 0elon 6439 | . . . . 5 ⊢ ∅ ∈ On | |
17 | 15, 16 | sselii 3991 | . . . 4 ⊢ ∅ ∈ ∪ (𝑅1 “ On) |
18 | eleq2 2827 | . . . 4 ⊢ (𝑊 = ∪ (𝑅1 “ On) → (∅ ∈ 𝑊 ↔ ∅ ∈ ∪ (𝑅1 “ On))) | |
19 | 17, 18 | mpbiri 258 | . . 3 ⊢ (𝑊 = ∪ (𝑅1 “ On) → ∅ ∈ 𝑊) |
20 | 4, 14, 19 | modelaxrep 44945 | . 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 1086 ∀wal 1534 = wceq 1536 ∈ wcel 2105 ∀wral 3058 ∃wrex 3067 ⊆ wss 3962 ∅c0 4338 ∪ cuni 4911 Tr wtr 5264 dom cdm 5688 ran crn 5689 “ cima 5691 Oncon0 6385 Fun wfun 6556 𝑅1cr1 9799 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-om 7887 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-en 8984 df-dom 8985 df-sdom 8986 df-r1 9801 df-rank 9802 |
This theorem is referenced by: (None) |
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