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Mirrors > Home > MPE Home > Th. List > tfr1a | Structured version Visualization version GIF version |
Description: A weak version of tfr1 8436 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 2735 | . . . 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 8422 | . . 3 ⊢ Fun recs(𝐺) |
3 | tfr.1 | . . . 4 ⊢ 𝐹 = recs(𝐺) | |
4 | 3 | funeqi 6589 | . . 3 ⊢ (Fun 𝐹 ↔ Fun recs(𝐺)) |
5 | 2, 4 | mpbir 231 | . 2 ⊢ Fun 𝐹 |
6 | 1 | tfrlem16 8432 | . . 3 ⊢ Lim dom recs(𝐺) |
7 | 3 | dmeqi 5918 | . . . 4 ⊢ dom 𝐹 = dom recs(𝐺) |
8 | limeq 6398 | . . . 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 231 | . 2 ⊢ Lim dom 𝐹 |
11 | 5, 10 | pm3.2i 470 | 1 ⊢ (Fun 𝐹 ∧ Lim dom 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1537 {cab 2712 ∀wral 3059 ∃wrex 3068 dom cdm 5689 ↾ cres 5691 Oncon0 6386 Lim wlim 6387 Fun wfun 6557 Fn wfn 6558 ‘cfv 6563 recscrecs 8409 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 |
This theorem is referenced by: tfr2b 8435 rdgfun 8455 rdgdmlim 8456 ordtypelem3 9558 ordtypelem4 9559 ordtypelem5 9560 ordtypelem6 9561 ordtypelem7 9562 ordtypelem9 9564 |
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