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Theorem madeun 33711
Description: The made set is the union of the old set and the new set. (Contributed by Scott Fenton, 9-Oct-2024.)
Assertion
Ref Expression
madeun ( M ‘𝐴) = (( O ‘𝐴) ∪ ( N ‘𝐴))

Proof of Theorem madeun
StepHypRef Expression
1 newval 33684 . . 3 ( N ‘𝐴) = (( M ‘𝐴) ∖ ( O ‘𝐴))
21uneq2i 4050 . 2 (( O ‘𝐴) ∪ ( N ‘𝐴)) = (( O ‘𝐴) ∪ (( M ‘𝐴) ∖ ( O ‘𝐴)))
3 oldssmade 33705 . . 3 ( O ‘𝐴) ⊆ ( M ‘𝐴)
4 undif 4371 . . 3 (( O ‘𝐴) ⊆ ( M ‘𝐴) ↔ (( O ‘𝐴) ∪ (( M ‘𝐴) ∖ ( O ‘𝐴))) = ( M ‘𝐴))
53, 4mpbi 233 . 2 (( O ‘𝐴) ∪ (( M ‘𝐴) ∖ ( O ‘𝐴))) = ( M ‘𝐴)
62, 5eqtr2i 2762 1 ( M ‘𝐴) = (( O ‘𝐴) ∪ ( N ‘𝐴))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  cdif 3840  cun 3841  wss 3843  cfv 6339   M cmade 33671   O cold 33672   N cnew 33673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7481
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-int 4837  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7129  df-ov 7175  df-oprab 7176  df-mpo 7177  df-wrecs 7978  df-recs 8039  df-1o 8133  df-2o 8134  df-no 33491  df-slt 33492  df-bday 33493  df-sslt 33621  df-scut 33623  df-made 33676  df-old 33677  df-new 33678
This theorem is referenced by:  oldsuc  33713
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