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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 7910 | . . . 4 ⊢ ∅ ∈ ω | |
| 2 | 1 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∅ ∈ ω) | 
| 3 | 1onn 8678 | . . . 4 ⊢ 1o ∈ ω | |
| 4 | 3 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 1o ∈ ω) | 
| 5 | simpl 482 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐴 ∈ 𝑉) | |
| 6 | simpr 484 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐵 ∈ 𝑊) | |
| 7 | 1n0 8526 | . . . . 5 ⊢ 1o ≠ ∅ | |
| 8 | 7 | necomi 2995 | . . . 4 ⊢ ∅ ≠ 1o | 
| 9 | 8 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∅ ≠ 1o) | 
| 10 | fnprg 6625 | . . 3 ⊢ (((∅ ∈ ω ∧ 1o ∈ ω) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ ∅ ≠ 1o) → {〈∅, 𝐴〉, 〈1o, 𝐵〉} Fn {∅, 1o}) | |
| 11 | 2, 4, 5, 6, 9, 10 | syl221anc 1383 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {〈∅, 𝐴〉, 〈1o, 𝐵〉} Fn {∅, 1o}) | 
| 12 | df2o3 8514 | . . 3 ⊢ 2o = {∅, 1o} | |
| 13 | 12 | fneq2i 6666 | . 2 ⊢ ({〈∅, 𝐴〉, 〈1o, 𝐵〉} Fn 2o ↔ {〈∅, 𝐴〉, 〈1o, 𝐵〉} Fn {∅, 1o}) | 
| 14 | 11, 13 | sylibr 234 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {〈∅, 𝐴〉, 〈1o, 𝐵〉} Fn 2o) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ≠ wne 2940 ∅c0 4333 {cpr 4628 〈cop 4632 Fn wfn 6556 ωcom 7887 1oc1o 8499 2oc2o 8500 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-fun 6563 df-fn 6564 df-om 7888 df-1o 8506 df-2o 8507 | 
| This theorem is referenced by: fnpr2ob 17603 xpsfeq 17608 xpsfrnel2 17609 xpsrnbas 17616 xpsaddlem 17618 xpsvsca 17622 xpsle 17624 xpstopnlem1 23817 xpstopnlem2 23819 xpsxmetlem 24389 xpsdsval 24391 xpsmet 24392 | 
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