rankuni2 - Metamath Proof Explorer

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Theorem rankuni2 9846

Description: The rank of a union. Part of Theorem 15.17(iv) of [Monk1] p. 112. (Contributed by NM, 30-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)

**Hypothesis**
Ref	Expression
ranksn.1	⊢ 𝐴 ∈ V

**Assertion**
Ref	Expression
rankuni2	⊢ (rank‘∪ 𝐴) = ∪ 𝑥 ∈ 𝐴 (rank‘𝑥)

Distinct variable group: 𝑥,𝐴

**Proof of Theorem rankuni2**
Step	Hyp	Ref	Expression
1		ranksn.1	. . 3 ⊢ 𝐴 ∈ V
2		unir1 9804	. . 3 ⊢ ∪ (𝑅₁ “ On) = V
3	1, 2	eleqtrri 2833	. 2 ⊢ 𝐴 ∈ ∪ (𝑅₁ “ On)
4		rankuni2b 9844	. 2 ⊢ (𝐴 ∈ ∪ (𝑅₁ “ On) → (rank‘∪ 𝐴) = ∪ 𝑥 ∈ 𝐴 (rank‘𝑥))
5	3, 4	ax-mp 5	1 ⊢ (rank‘∪ 𝐴) = ∪ 𝑥 ∈ 𝐴 (rank‘𝑥)

Colors of variables: wff setvar class

Syntax hints: = wceq 1542 ∈ wcel 2107 Vcvv 3475 ∪ cuni 4907 ∪ ciun 4996 “ cima 5678 Oncon0 6361 ‘cfv 6540 𝑅₁cr1 9753 rankcrnk 9754

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-reg 9583 ax-inf2 9632

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7407 df-om 7851 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-r1 9755 df-rank 9756

This theorem is referenced by: rankuni 9854 rankbnd2 9860

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