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| Description: A weak version of tfr1 8438 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 2736 | . . . 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 8424 | . . 3 ⊢ Fun recs(𝐺) | 
| 3 | tfr.1 | . . . 4 ⊢ 𝐹 = recs(𝐺) | |
| 4 | 3 | funeqi 6586 | . . 3 ⊢ (Fun 𝐹 ↔ Fun recs(𝐺)) | 
| 5 | 2, 4 | mpbir 231 | . 2 ⊢ Fun 𝐹 | 
| 6 | 1 | tfrlem16 8434 | . . 3 ⊢ Lim dom recs(𝐺) | 
| 7 | 3 | dmeqi 5914 | . . . 4 ⊢ dom 𝐹 = dom recs(𝐺) | 
| 8 | limeq 6395 | . . . 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 1539 {cab 2713 ∀wral 3060 ∃wrex 3069 dom cdm 5684 ↾ cres 5686 Oncon0 6383 Lim wlim 6384 Fun wfun 6554 Fn wfn 6555 ‘cfv 6560 recscrecs 8411 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 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-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 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 | 
| This theorem is referenced by: tfr2b 8437 rdgfun 8457 rdgdmlim 8458 ordtypelem3 9561 ordtypelem4 9562 ordtypelem5 9563 ordtypelem6 9564 ordtypelem7 9565 ordtypelem9 9567 | 
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