![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > tfr1a | Structured version Visualization version GIF version |
Description: A weak version of tfr1 8411 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 2728 | . . . 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 8397 | . . 3 ⊢ Fun recs(𝐺) |
3 | tfr.1 | . . . 4 ⊢ 𝐹 = recs(𝐺) | |
4 | 3 | funeqi 6568 | . . 3 ⊢ (Fun 𝐹 ↔ Fun recs(𝐺)) |
5 | 2, 4 | mpbir 230 | . 2 ⊢ Fun 𝐹 |
6 | 1 | tfrlem16 8407 | . . 3 ⊢ Lim dom recs(𝐺) |
7 | 3 | dmeqi 5901 | . . . 4 ⊢ dom 𝐹 = dom recs(𝐺) |
8 | limeq 6375 | . . . 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 470 | 1 ⊢ (Fun 𝐹 ∧ Lim dom 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 = wceq 1534 {cab 2705 ∀wral 3057 ∃wrex 3066 dom cdm 5672 ↾ cres 5674 Oncon0 6363 Lim wlim 6364 Fun wfun 6536 Fn wfn 6537 ‘cfv 6542 recscrecs 8384 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 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 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 |
This theorem is referenced by: tfr2b 8410 rdgfun 8430 rdgdmlim 8431 ordtypelem3 9537 ordtypelem4 9538 ordtypelem5 9539 ordtypelem6 9540 ordtypelem7 9541 ordtypelem9 9543 |
Copyright terms: Public domain | W3C validator |