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Mirrors > Home > MPE Home > Th. List > fpr2a | Structured version Visualization version GIF version |
Description: Weak version of fpr2 8112 which is useful for proofs that avoid the axiom of replacement. (Contributed by Scott Fenton, 18-Nov-2024.) |
Ref | Expression |
---|---|
fpr2a.1 | ⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺) |
Ref | Expression |
---|---|
fpr2a | ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑋 ∈ dom 𝐹) → (𝐹‘𝑋) = (𝑋𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6771 | . . . . 5 ⊢ (𝑦 = 𝑋 → (𝐹‘𝑦) = (𝐹‘𝑋)) | |
2 | id 22 | . . . . . 6 ⊢ (𝑦 = 𝑋 → 𝑦 = 𝑋) | |
3 | predeq3 6205 | . . . . . . 7 ⊢ (𝑦 = 𝑋 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑋)) | |
4 | 3 | reseq2d 5890 | . . . . . 6 ⊢ (𝑦 = 𝑋 → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))) |
5 | 2, 4 | oveq12d 7290 | . . . . 5 ⊢ (𝑦 = 𝑋 → (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝑋𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)))) |
6 | 1, 5 | eqeq12d 2756 | . . . 4 ⊢ (𝑦 = 𝑋 → ((𝐹‘𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝐹‘𝑋) = (𝑋𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))) |
7 | 6 | imbi2d 341 | . . 3 ⊢ (𝑦 = 𝑋 → (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) → (𝐹‘𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ ((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) → (𝐹‘𝑋) = (𝑋𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)))))) |
8 | eqid 2740 | . . . . 5 ⊢ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} | |
9 | fpr2a.1 | . . . . 5 ⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺) | |
10 | 8, 9 | fprlem1 8108 | . . . . 5 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} ∧ ℎ ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))})) → ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 𝑢 = 𝑣)) |
11 | 8, 9, 10 | frrlem10 8103 | . . . 4 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑦 ∈ dom 𝐹) → (𝐹‘𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) |
12 | 11 | expcom 414 | . . 3 ⊢ (𝑦 ∈ dom 𝐹 → ((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) → (𝐹‘𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))) |
13 | 7, 12 | vtoclga 3512 | . 2 ⊢ (𝑋 ∈ dom 𝐹 → ((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) → (𝐹‘𝑋) = (𝑋𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))) |
14 | 13 | impcom 408 | 1 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑋 ∈ dom 𝐹) → (𝐹‘𝑋) = (𝑋𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1542 ∃wex 1786 ∈ wcel 2110 {cab 2717 ∀wral 3066 ⊆ wss 3892 Po wpo 5502 Fr wfr 5542 Se wse 5543 dom cdm 5590 ↾ cres 5592 Predcpred 6200 Fn wfn 6427 ‘cfv 6432 (class class class)co 7272 frecscfrecs 8088 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pr 5356 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5490 df-po 5504 df-fr 5545 df-se 5546 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-iota 6390 df-fun 6434 df-fn 6435 df-fv 6440 df-ov 7275 df-frecs 8089 |
This theorem is referenced by: fpr2 8112 wfr2a 8157 |
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