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Mirrors > Home > MPE Home > Th. List > inf3lemc | Structured version Visualization version GIF version |
Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8890 for detailed description. (Contributed by NM, 28-Oct-1996.) |
Ref | Expression |
---|---|
inf3lem.1 | ⊢ 𝐺 = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}) |
inf3lem.2 | ⊢ 𝐹 = (rec(𝐺, ∅) ↾ ω) |
inf3lem.3 | ⊢ 𝐴 ∈ V |
inf3lem.4 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
inf3lemc | ⊢ (𝐴 ∈ ω → (𝐹‘suc 𝐴) = (𝐺‘(𝐹‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frsuc 7874 | . 2 ⊢ (𝐴 ∈ ω → ((rec(𝐺, ∅) ↾ ω)‘suc 𝐴) = (𝐺‘((rec(𝐺, ∅) ↾ ω)‘𝐴))) | |
2 | inf3lem.2 | . . 3 ⊢ 𝐹 = (rec(𝐺, ∅) ↾ ω) | |
3 | 2 | fveq1i 6497 | . 2 ⊢ (𝐹‘suc 𝐴) = ((rec(𝐺, ∅) ↾ ω)‘suc 𝐴) |
4 | 2 | fveq1i 6497 | . . 3 ⊢ (𝐹‘𝐴) = ((rec(𝐺, ∅) ↾ ω)‘𝐴) |
5 | 4 | fveq2i 6499 | . 2 ⊢ (𝐺‘(𝐹‘𝐴)) = (𝐺‘((rec(𝐺, ∅) ↾ ω)‘𝐴)) |
6 | 1, 3, 5 | 3eqtr4g 2833 | 1 ⊢ (𝐴 ∈ ω → (𝐹‘suc 𝐴) = (𝐺‘(𝐹‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2050 {crab 3086 Vcvv 3409 ∩ cin 3822 ⊆ wss 3823 ∅c0 4172 ↦ cmpt 5004 ↾ cres 5405 suc csuc 6028 ‘cfv 6185 ωcom 7394 reccrdg 7847 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-om 7395 df-wrecs 7748 df-recs 7810 df-rdg 7848 |
This theorem is referenced by: inf3lemd 8882 inf3lem1 8883 inf3lem2 8884 inf3lem3 8885 |
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