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Mirrors > Home > MPE Home > Th. List > tfr1a | Structured version Visualization version GIF version |
Description: A weak version of tfr1 7654 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 2752 | . . . 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 7640 | . . 3 ⊢ Fun recs(𝐺) |
3 | tfr.1 | . . . 4 ⊢ 𝐹 = recs(𝐺) | |
4 | 3 | funeqi 6062 | . . 3 ⊢ (Fun 𝐹 ↔ Fun recs(𝐺)) |
5 | 2, 4 | mpbir 221 | . 2 ⊢ Fun 𝐹 |
6 | 1 | tfrlem16 7650 | . . 3 ⊢ Lim dom recs(𝐺) |
7 | 3 | dmeqi 5472 | . . . 4 ⊢ dom 𝐹 = dom recs(𝐺) |
8 | limeq 5888 | . . . 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 221 | . 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 196 ∧ wa 383 = wceq 1624 {cab 2738 ∀wral 3042 ∃wrex 3043 dom cdm 5258 ↾ cres 5260 Oncon0 5876 Lim wlim 5877 Fun wfun 6035 Fn wfn 6036 ‘cfv 6041 recscrecs 7628 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1863 ax-4 1878 ax-5 1980 ax-6 2046 ax-7 2082 ax-8 2133 ax-9 2140 ax-10 2160 ax-11 2175 ax-12 2188 ax-13 2383 ax-ext 2732 ax-sep 4925 ax-nul 4933 ax-pow 4984 ax-pr 5047 ax-un 7106 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1627 df-ex 1846 df-nf 1851 df-sb 2039 df-eu 2603 df-mo 2604 df-clab 2739 df-cleq 2745 df-clel 2748 df-nfc 2883 df-ne 2925 df-ral 3047 df-rex 3048 df-reu 3049 df-rab 3051 df-v 3334 df-sbc 3569 df-csb 3667 df-dif 3710 df-un 3712 df-in 3714 df-ss 3721 df-pss 3723 df-nul 4051 df-if 4223 df-pw 4296 df-sn 4314 df-pr 4316 df-tp 4318 df-op 4320 df-uni 4581 df-iun 4666 df-br 4797 df-opab 4857 df-mpt 4874 df-tr 4897 df-id 5166 df-eprel 5171 df-po 5179 df-so 5180 df-fr 5217 df-we 5219 df-xp 5264 df-rel 5265 df-cnv 5266 df-co 5267 df-dm 5268 df-rn 5269 df-res 5270 df-ima 5271 df-pred 5833 df-ord 5879 df-on 5880 df-lim 5881 df-suc 5882 df-iota 6004 df-fun 6043 df-fn 6044 df-f 6045 df-f1 6046 df-fo 6047 df-f1o 6048 df-fv 6049 df-wrecs 7568 df-recs 7629 |
This theorem is referenced by: tfr2b 7653 rdgfun 7673 rdgdmlim 7674 ordtypelem3 8582 ordtypelem4 8583 ordtypelem5 8584 ordtypelem6 8585 ordtypelem7 8586 ordtypelem9 8588 |
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