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Mirrors > Home > ILE Home > Th. List > tfrlem3-2d | GIF version |
Description: Lemma for transfinite recursion which changes a bound variable (Contributed by Jim Kingdon, 2-Jul-2019.) |
Ref | Expression |
---|---|
tfrlem3-2d.1 | ⊢ (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹‘𝑥) ∈ V)) |
Ref | Expression |
---|---|
tfrlem3-2d | ⊢ (𝜑 → (Fun 𝐹 ∧ (𝐹‘𝑔) ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tfrlem3-2d.1 | . . 3 ⊢ (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹‘𝑥) ∈ V)) | |
2 | fveq2 5527 | . . . . . 6 ⊢ (𝑥 = 𝑔 → (𝐹‘𝑥) = (𝐹‘𝑔)) | |
3 | 2 | eleq1d 2256 | . . . . 5 ⊢ (𝑥 = 𝑔 → ((𝐹‘𝑥) ∈ V ↔ (𝐹‘𝑔) ∈ V)) |
4 | 3 | anbi2d 464 | . . . 4 ⊢ (𝑥 = 𝑔 → ((Fun 𝐹 ∧ (𝐹‘𝑥) ∈ V) ↔ (Fun 𝐹 ∧ (𝐹‘𝑔) ∈ V))) |
5 | 4 | cbvalv 1927 | . . 3 ⊢ (∀𝑥(Fun 𝐹 ∧ (𝐹‘𝑥) ∈ V) ↔ ∀𝑔(Fun 𝐹 ∧ (𝐹‘𝑔) ∈ V)) |
6 | 1, 5 | sylib 122 | . 2 ⊢ (𝜑 → ∀𝑔(Fun 𝐹 ∧ (𝐹‘𝑔) ∈ V)) |
7 | 6 | 19.21bi 1568 | 1 ⊢ (𝜑 → (Fun 𝐹 ∧ (𝐹‘𝑔) ∈ V)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∀wal 1361 ∈ wcel 2158 Vcvv 2749 Fun wfun 5222 ‘cfv 5228 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-rex 2471 df-v 2751 df-un 3145 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-br 4016 df-iota 5190 df-fv 5236 |
This theorem is referenced by: tfrlemisucfn 6339 tfrlemisucaccv 6340 tfrlemibxssdm 6342 tfrlemibfn 6343 tfrlemi14d 6348 |
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