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Mirrors > Home > MPE Home > Th. List > tfr1a | Structured version Visualization version GIF version |
Description: A weak version of tfr1 8032 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 2821 | . . . 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 8018 | . . 3 ⊢ Fun recs(𝐺) |
3 | tfr.1 | . . . 4 ⊢ 𝐹 = recs(𝐺) | |
4 | 3 | funeqi 6375 | . . 3 ⊢ (Fun 𝐹 ↔ Fun recs(𝐺)) |
5 | 2, 4 | mpbir 233 | . 2 ⊢ Fun 𝐹 |
6 | 1 | tfrlem16 8028 | . . 3 ⊢ Lim dom recs(𝐺) |
7 | 3 | dmeqi 5772 | . . . 4 ⊢ dom 𝐹 = dom recs(𝐺) |
8 | limeq 6202 | . . . 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 233 | . 2 ⊢ Lim dom 𝐹 |
11 | 5, 10 | pm3.2i 473 | 1 ⊢ (Fun 𝐹 ∧ Lim dom 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1533 {cab 2799 ∀wral 3138 ∃wrex 3139 dom cdm 5554 ↾ cres 5556 Oncon0 6190 Lim wlim 6191 Fun wfun 6348 Fn wfn 6349 ‘cfv 6354 recscrecs 8006 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-wrecs 7946 df-recs 8007 |
This theorem is referenced by: tfr2b 8031 rdgfun 8051 rdgdmlim 8052 ordtypelem3 8983 ordtypelem4 8984 ordtypelem5 8985 ordtypelem6 8986 ordtypelem7 8987 ordtypelem9 8989 |
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