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Mirrors > Home > MPE Home > Th. List > tfr1a | Structured version Visualization version GIF version |
Description: A weak version of tfr1 8384 which is useful for proofs that avoid the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.) |
Ref | Expression |
---|---|
tfr.1 | ⊢ 𝐹 = recs(𝐺) |
Ref | Expression |
---|---|
tfr1a | ⊢ (Fun 𝐹 ∧ Lim dom 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . . 4 ⊢ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))} | |
2 | 1 | tfrlem7 8370 | . . 3 ⊢ Fun recs(𝐺) |
3 | tfr.1 | . . . 4 ⊢ 𝐹 = recs(𝐺) | |
4 | 3 | funeqi 6561 | . . 3 ⊢ (Fun 𝐹 ↔ Fun recs(𝐺)) |
5 | 2, 4 | mpbir 230 | . 2 ⊢ Fun 𝐹 |
6 | 1 | tfrlem16 8380 | . . 3 ⊢ Lim dom recs(𝐺) |
7 | 3 | dmeqi 5899 | . . . 4 ⊢ dom 𝐹 = dom recs(𝐺) |
8 | limeq 6368 | . . . 4 ⊢ (dom 𝐹 = dom recs(𝐺) → (Lim dom 𝐹 ↔ Lim dom recs(𝐺))) | |
9 | 7, 8 | ax-mp 5 | . . 3 ⊢ (Lim dom 𝐹 ↔ Lim dom recs(𝐺)) |
10 | 6, 9 | mpbir 230 | . 2 ⊢ Lim dom 𝐹 |
11 | 5, 10 | pm3.2i 472 | 1 ⊢ (Fun 𝐹 ∧ Lim dom 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 = wceq 1542 {cab 2710 ∀wral 3062 ∃wrex 3071 dom cdm 5672 ↾ cres 5674 Oncon0 6356 Lim wlim 6357 Fun wfun 6529 Fn wfn 6530 ‘cfv 6535 recscrecs 8357 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pr 5423 ax-un 7712 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-ov 7399 df-2nd 7963 df-frecs 8253 df-wrecs 8284 df-recs 8358 |
This theorem is referenced by: tfr2b 8383 rdgfun 8403 rdgdmlim 8404 ordtypelem3 9502 ordtypelem4 9503 ordtypelem5 9504 ordtypelem6 9505 ordtypelem7 9506 ordtypelem9 9508 |
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