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Mirrors > Home > MPE Home > Th. List > fnpr2o | Structured version Visualization version GIF version |
Description: Function with a domain of 2o. (Contributed by Jim Kingdon, 25-Sep-2023.) |
Ref | Expression |
---|---|
fnpr2o | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {〈∅, 𝐴〉, 〈1o, 𝐵〉} Fn 2o) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano1 7598 | . . . 4 ⊢ ∅ ∈ ω | |
2 | 1 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∅ ∈ ω) |
3 | 1onn 8262 | . . . 4 ⊢ 1o ∈ ω | |
4 | 3 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 1o ∈ ω) |
5 | simpl 485 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐴 ∈ 𝑉) | |
6 | simpr 487 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐵 ∈ 𝑊) | |
7 | 1n0 8116 | . . . . 5 ⊢ 1o ≠ ∅ | |
8 | 7 | necomi 3069 | . . . 4 ⊢ ∅ ≠ 1o |
9 | 8 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∅ ≠ 1o) |
10 | fnprg 6410 | . . 3 ⊢ (((∅ ∈ ω ∧ 1o ∈ ω) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ ∅ ≠ 1o) → {〈∅, 𝐴〉, 〈1o, 𝐵〉} Fn {∅, 1o}) | |
11 | 2, 4, 5, 6, 9, 10 | syl221anc 1376 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {〈∅, 𝐴〉, 〈1o, 𝐵〉} Fn {∅, 1o}) |
12 | df2o3 8114 | . . 3 ⊢ 2o = {∅, 1o} | |
13 | 12 | fneq2i 6448 | . 2 ⊢ ({〈∅, 𝐴〉, 〈1o, 𝐵〉} Fn 2o ↔ {〈∅, 𝐴〉, 〈1o, 𝐵〉} Fn {∅, 1o}) |
14 | 11, 13 | sylibr 236 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {〈∅, 𝐴〉, 〈1o, 𝐵〉} Fn 2o) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2113 ≠ wne 3015 ∅c0 4288 {cpr 4566 〈cop 4570 Fn wfn 6347 ωcom 7577 1oc1o 8092 2oc2o 8093 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5200 ax-nul 5207 ax-pr 5327 ax-un 7458 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3495 df-sbc 3771 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4465 df-pw 4538 df-sn 4565 df-pr 4567 df-tp 4569 df-op 4571 df-uni 4836 df-br 5064 df-opab 5126 df-tr 5170 df-id 5457 df-eprel 5462 df-po 5471 df-so 5472 df-fr 5511 df-we 5513 df-xp 5558 df-rel 5559 df-cnv 5560 df-co 5561 df-dm 5562 df-ord 6191 df-on 6192 df-lim 6193 df-suc 6194 df-fun 6354 df-fn 6355 df-om 7578 df-1o 8099 df-2o 8100 |
This theorem is referenced by: fnpr2ob 16827 xpsfeq 16832 xpsfrnel2 16833 xpsrnbas 16840 xpsaddlem 16842 xpsvsca 16846 xpsle 16848 xpstopnlem1 22413 xpstopnlem2 22415 xpsxmetlem 22985 xpsdsval 22987 xpsmet 22988 |
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