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Mirrors > Home > MPE Home > Th. List > xpscf | Structured version Visualization version GIF version |
Description: Equivalent condition for the pair function to be a proper function on 𝐴. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xpscf | ⊢ (◡({𝑋} +𝑐 {𝑌}):2𝑜⟶𝐴 ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ifid 4075 | . . . . . 6 ⊢ if(𝑘 = ∅, 𝐴, 𝐴) = 𝐴 | |
2 | 1 | eleq2i 2680 | . . . . 5 ⊢ ((◡({𝑋} +𝑐 {𝑌})‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐴) ↔ (◡({𝑋} +𝑐 {𝑌})‘𝑘) ∈ 𝐴) |
3 | 2 | ralbii 2963 | . . . 4 ⊢ (∀𝑘 ∈ 2𝑜 (◡({𝑋} +𝑐 {𝑌})‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐴) ↔ ∀𝑘 ∈ 2𝑜 (◡({𝑋} +𝑐 {𝑌})‘𝑘) ∈ 𝐴) |
4 | 3 | anbi2i 726 | . . 3 ⊢ ((◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜 ∧ ∀𝑘 ∈ 2𝑜 (◡({𝑋} +𝑐 {𝑌})‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐴)) ↔ (◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜 ∧ ∀𝑘 ∈ 2𝑜 (◡({𝑋} +𝑐 {𝑌})‘𝑘) ∈ 𝐴)) |
5 | ovex 6577 | . . . . 5 ⊢ ({𝑋} +𝑐 {𝑌}) ∈ V | |
6 | 5 | cnvex 7006 | . . . 4 ⊢ ◡({𝑋} +𝑐 {𝑌}) ∈ V |
7 | 6 | elixp 7801 | . . 3 ⊢ (◡({𝑋} +𝑐 {𝑌}) ∈ X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐴) ↔ (◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜 ∧ ∀𝑘 ∈ 2𝑜 (◡({𝑋} +𝑐 {𝑌})‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐴))) |
8 | ffnfv 6295 | . . 3 ⊢ (◡({𝑋} +𝑐 {𝑌}):2𝑜⟶𝐴 ↔ (◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜 ∧ ∀𝑘 ∈ 2𝑜 (◡({𝑋} +𝑐 {𝑌})‘𝑘) ∈ 𝐴)) | |
9 | 4, 7, 8 | 3bitr4i 291 | . 2 ⊢ (◡({𝑋} +𝑐 {𝑌}) ∈ X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐴) ↔ ◡({𝑋} +𝑐 {𝑌}):2𝑜⟶𝐴) |
10 | xpsfrnel2 16048 | . 2 ⊢ (◡({𝑋} +𝑐 {𝑌}) ∈ X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐴) ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) | |
11 | 9, 10 | bitr3i 265 | 1 ⊢ (◡({𝑋} +𝑐 {𝑌}):2𝑜⟶𝐴 ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∅c0 3874 ifcif 4036 {csn 4125 ◡ccnv 5037 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 2𝑜c2o 7441 Xcixp 7794 +𝑐 ccda 8872 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-cda 8873 |
This theorem is referenced by: xpsmnd 17153 xpsgrp 17357 dmdprdpr 18271 dprdpr 18272 xpstopnlem1 21422 xpstps 21423 xpsxms 22149 xpsms 22150 |
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